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Analysis 101:
Curves and Length

Garth Warner
Department of Mathematics
University of Washington

ABSTRACT

This is a systematic accounting of the classical theorems of Jordan and Tonelli, as well as an introduction to the theory of the Weierstrass integral which in its definitive form is due to Cesari. This is installment II of a four part discussion of certain aspects of Real Analysis: Functions of a Single Variable, Curves and Length, Functions of Several Variables, and Surfaces and Area.

ACKNOWLEDGEMENT

Many thanks to David Clark for his rendering the original transcript into AMS-LaTeX. Both of us also thank Judith Clare for her meticulous proofreading.

CURVES AND LENGTH

§1. FUNDAMENTALS

§2. ESTIMATES

§3. EQUIVALENCES

§4. FRÉCHET DISTANCE

§5. THE REPRESENTATION THEOREM

§6. INDUCED MEASURES

§7. TWO THEOREMS

§8. LINE INTEGRALS

§9. QUASI ADDITIVITY

§10. LINE INTEGRALS (bis)

§11. EXAMPLES

REFERENCES

$\boldsymbol{\S}$ 1. FUNDAMENTALS

myn1

NOTATION Given

\underline{x}=(x_{1},\ldots,x_{M})\in\mathbb{R}^{M}\quad(M=1,2,\ldots),

put

\left\lVert\underline{x}\right\rVert=(x_{1}^{2}+\ldots+x_{M}^{2})^{\frac{1}{2}},

hence

\left|x_{m}\right|\ \leq\ \left\lVert\underline{x}\right\rVert\ \leq\ \left|x_{1}\right|+\ldots+\left|x_{M}\right|\quad(m=1,\ldots,M).

myn2

DEFINITION A function $\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}$ is said to be a curve $C$ , denoted $C\longleftrightarrow\underline{f}$ , where

\underline{f}(x)=(f_{1}(x),\ldots,f_{M}(x))\quad(a\leq x\leq b).

myn3

EXAMPLE Every function $f:[a,b]\rightarrow\mathbb{R}$ gives rise to a curve $C$ in $\mathbb{R}^{2}$ , viz. the arrow

x\rightarrow(x,f(x)).

myn4

DEFINITION The graph of $C$ , denoted $[C]$ , is the range of $\underline{f}$ .

myn5

EXAMPLE Take $M=2$ , let $k=1,2,\ldots$ , and put

\underline{f_{k}}(x)\ =\ \left(\sin^{2}(kx),0\right)\qquad\left(0\leq x\leq\frac{\pi}{2}\right).

Then the $\underline{f_{k}}$ all have the same range, i.e.,

[C_{1}]\ =\ [C_{2}]\ =\ \ldots\qquad\text{ if }\quad C_{k}\longleftrightarrow\underline{f_{k}}

but the $C_{k}$ are different curves.

myn6

REMARK If $C$ is a continuous curve, then its graph $[C]$ is closed, bounded, connected, and uniformly locally connected. Owing to a theorem of Hahn and Mazurkiewicz, these properties are characteristic: Any such set is the graph of a continuous curve. So, e.g., a square in $\mathbb{R}^{2}$ is the graph of a continuous curve, a cube in $\mathbb{R}^{3}$ is the graph of a continuous curve etc.

myn7

DEFINITION The length of a curve $C$ , denoted $\ell(C)$ , is

\text{T}_{\underline{f}}[a,b]\ \equiv\ \sup\limits_{P\in\mathcal{P}[a,b]}\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert,

$C$ being termed rectifiable if $\ell(C)<+\infty$ .

[Note: If $C$ is continuous and rectifiable, then $\forall\ \varepsilon>0,\ \exists\ \delta>0:$

\left\lVert P\right\rVert<\delta\implies\bigvee\limits_{a}^{b}\ (\underline{f};P)\ \equiv\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\ >\ \ell(C)-\varepsilon.]

myn8

LEMMA Given a curve $C$ ,

\text{T}_{f_{m}}[a,b]\ \leq\ \ell(C)\ \leq\ \text{T}_{f_{1}}[a,b]+\ldots+\text{T}_{f_{M}}[a,b]\qquad(1\leq m\leq M).

myn9

SCHOLIUM $C$ is rectifiable iff

f_{1}\in\text{BV}[a,b],\ldots,f_{M}\in\text{BV}[a,b].

myn10

THEOREM Let

\begin{cases}\ C_{n}\longleftrightarrow\underline{f_{n}}:[a,b]\rightarrow\mathbb{R}^{M}\\ \ C\hskip 4.26773pt\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}\\ \end{cases}

and assume that $\underline{f_{n}}$ converges pointwise to $\underline{f}$ $-$ then

\ell(C)\ \leq\ \liminf\limits_{n\rightarrow\infty}\ \ell(C_{n}).

A continuous curve

\Gamma\longleftrightarrow\gamma:[a,b]\rightarrow\mathbb{R}^{M}

is said to be a polygonal line (and $\gamma$ quasi linear in $[a,b]$ ) if there exists a $P\in\mathcal{P}[a,b]$ in each segment of which $\underline{\gamma}$ is linear or a constant.

myn11

DEFINITION The elementary length $\ell_{e}(\Gamma)$ of $\Gamma$ is the sum of the lengths of these segments, hence $\ell_{e}(\Gamma)=\ell(\Gamma)$ .

myn12

NOTATION Given a continuous curve $C$ , denote by $\Gamma(C)$ the set of all sequences

\Gamma_{n}\longleftrightarrow\underline{\gamma_{n}}:[a,b]\rightarrow\mathbb{R}^{M}

of polygonal lines such that

\gamma_{n}\rightarrow\underline{f}\qquad(n\rightarrow\infty)

uniformly in $[a,b]$ .

Therefore

\ell(C)\ \leq\ \liminf\limits_{n\rightarrow\infty}\ \ell(\Gamma_{n})\ =\ \liminf\limits_{n\rightarrow\infty}\ \ell_{e}(\Gamma_{n}).

On the other hand, by definition, there is some $\{\Gamma_{n}\}\in\Gamma(C)$ such that

\ell_{e}(\Gamma_{n})\rightarrow\ell(C)\qquad(n\rightarrow\infty).

myn13

SCHOLIUM If $C$ is a continuous curve, then

\ell(C)\ =\ \inf\limits_{\{\Gamma_{n}\}\in\Gamma(C)}\ [\liminf\limits_{n\rightarrow\infty}\ell_{e}(\Gamma_{n})].

myn14

REMARK Let

C\longleftrightarrow f:[a,b]\rightarrow\mathbb{R}^{M}.

Assume: $C$ is continuous and rectifiable $-$ then $f$ can be decomposed as a sum

f=f_{AC}+f_{C}

where $f_{AC}$ is absolutely continuous and $f_{C}$ is continuous and singular.

Therefore

\ell(C)\ =\ \text{T}_{f_{AC}}[a,b]+\text{T}_{f_{C}}[a,b].

$\boldsymbol{\S}$ 2. ESTIMATES

myn1

NOTATION Write

\text{T}_{\underline{f}}[a,b]

in place of $\ell(C)$ .

myn2

DEFINITION Assume that $C$ is rectifiable $-$ then the arc length function

s:[a,b]\rightarrow\mathbb{R}

is defined by the prescription

s(x)=\text{T}_{\underline{f}}[a,x]\qquad(a\leq x\leq b).

Obviously

s(a)\ =\ 0,\quad s(b)\ =\ \ell(C),

and $s$ is an increasing function.

myn3

LEMMA If $C$ is continuous and rectifiable, then $s$ is continuous as are the $\text{T}_{f_{m}}[a,-]\quad(m=1,\ldots,M)$ .

myn4

LEMMA If $C$ is continuous and rectifiable, then $s$ is absolutely continuous iff all the $\text{T}_{f_{m}}[a,-]\quad(m=1,\ldots,M)$ are absolutely continuous, hence iff all the $f_{m}\quad(m=1,\ldots,M)$ are absolutely continuous.

If $C$ is continuous and rectifiable, then the $f_{m}\in\text{BV}[a,b]$ , thus the derivatives $f_{m}^{\prime}$ exist almost everywhere in $[a,b]$ and are Lebesgue integrable. On the other hand, $s$ is an increasing function, thus it too is differentiable almost everywhere in $[a,b]$ and is Lebesgue integrable.

myn5

SUBLEMMA The connection between $\underline{f}^{\prime}$ and $s^{\prime}$ is given by the relation

\left\lVert\underline{f}^{\prime}\right\rVert\ \leq\ s^{\prime}

almost everywhere in $[a,b]$ .

[For any subinterval $[\alpha,\beta]\subset[a,b]$ ,

\left\lVert\underline{f}(\beta)-\underline{f}(\alpha)\right\rVert\ \leq\ s(\beta)-s(\alpha).]

myn6

LEMMA

\ell(C)\ =\ s(b)-s(a)\ \geq\ \int\limits_{a}^{b}s^{\prime}\ \geq\ \int\limits_{a}^{b}\left\lVert\underline{f}^{\prime}\right\rVert.

I.e.: Under the assumption that $C$ is continuous and rectifiable,

\ell(C)\ \geq\ \int\limits_{a}^{b}\ \left\lVert\underline{f}^{\prime}\right\rVert.

myn7

THEOREM

\ell(C)\ =\ \int\limits_{a}^{b}\ \left\lVert\underline{f}^{\prime}\right\rVert

iff all the $f_{m}$ $(m=1,\ldots,M)$ are absolutely continuous.

This is established in the discussion to follow.

• Suppose that the equality sign obtains, hence

s(b)-s(a)\ =\ \int\limits_{a}^{b}\ s^{\prime}.

But also

	$\displaystyle s(x)-s(a)\ \geq\ \int\limits_{a}^{x}\ s^{\prime}$
	$\displaystyle s(b)-s(x)\ \geq\ \int\limits_{x}^{b}s^{\prime}.$

	$\displaystyle s(x)-s(a)\ >\ \int\limits_{a}^{x}s^{\prime}$
	$\displaystyle s(b)-s(x)\ \geq\ \int\limits_{x}^{b}s^{\prime},$

then

s(b)-s(a)\ >\ \int\limits_{a}^{b}s^{\prime},

a contradiction. Therefore

s(x)-s(a)\ =\ \int\limits_{a}^{x}s^{\prime}

$\implies\qquad s\in\text{AC}[a,b]$

$\implies\qquad f_{m}\in\text{AC}[a,b]\quad(m=1,\ldots,M).$

• Consider the other direction, i.e., assume that the $f_{m}\in\text{AC}[a,b]$ , the claim being that

\ell(C)=\int\limits_{a}^{b}\left\lVert f^{\prime}\right\rVert.

Given $P\in\mathcal{P}[a,b]$ , write

	$\displaystyle\sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\$	$\displaystyle=\ \sum\limits_{i=1}^{n}\ \left[\sum\limits_{m=1}^{M}(f_{m}(x_{i})-f_{m}(x_{i-1}))^{2}\right]^{\frac{1}{2}}$
		$\displaystyle=\ \sum\limits_{i=1}^{n}\ \left[\sum\limits_{m=1}^{M}\ \bigg{(}\hskip 1.42271pt\int\limits_{x_{i}}^{x_{i}}\ f_{m}^{\prime}\bigg{)}^{2}\right]^{\frac{1}{2}}$
		$\displaystyle\leq\ \sum\limits_{i=1}^{n}\ \int\limits_{x_{i}}^{x_{i}}\ \bigg{(}\hskip 1.42271pt\sum\limits_{m=1}^{M}(f_{m}^{\prime})^{2}\bigg{)}^{1/2}$
		$\displaystyle=\ \int\limits_{a}^{b}\ \left\lVert f^{\prime}\right\rVert.$

Taking the sup of the first term over all $P$ then gives

\ell(C)\ \leq\ \int\limits_{a}^{b}\ \left\lVert\underline{f}^{\prime}\right\rVert\quad(\leq\ell(C))

$\implies$

\ell(C)\ =\ \int\limits_{a}^{b}\ \left\lVert\underline{f}^{\prime}\right\rVert.

myn8

N.B. Under canonical assumptions,

\bigg{(}\hskip 1.42271pt\bigg{(}\hskip 1.42271pt\int\limits_{\mathbb{X}}\phi_{1}\bigg{)}^{2}+\ldots+\bigg{(}\int\limits_{\mathbb{X}}\ \phi_{n}\bigg{)}^{2}\hskip 1.42271pt\bigg{)}^{\frac{1}{2}}\ \leq\ \int\limits_{\mathbb{X}}\left(\phi_{1}^{2}+\ldots+\phi_{n}^{2}\right)^{\frac{1}{2}}.

myn9

RAPPEL Suppose that $f\in\text{BV}[a,b]$ $-$ then for almost all $x\in[a,b]$ ,

\left|f^{\prime}(x)\right|\ =\ \text{T}_{f^{\prime}}[a,x].

myn10

LEMMA Suppose that $C$ is continuous and rectifiable $-$ then

s^{\prime}\ =\ \left\lVert\underline{f}^{\prime}\right\rVert

almost everywhere in $[a,b]$ .

PROOF Since

\left\lVert\underline{f}^{\prime}\right\rVert\ \leq\ s^{\prime},

it suffices to show that

s^{\prime}\ \leq\ \left\lVert\underline{f}^{\prime}\right\rVert.

Let $E_{0}\subset[a,b]$ be the set of $x$ such that $\underline{f}$ and $s$ are differentiable at $x$ and $s^{\prime}(x)>\left\lVert\underline{f}^{\prime}(x)\right\rVert$ and for $k=1,2,\ldots$ , let $E_{k}$ be the set of $x\in E_{0}$ such that

\frac{s(t_{2})-s(t_{1})}{t_{2}-t_{1}}\ \geq\ \frac{\left\lVert\underline{f}(t_{2})-\underline{f}(t_{1})\right\rVert}{t_{2}-t_{1}}+\frac{1}{k}

for all intervals $[t_{1},t_{2}]$ such that $x\in[t_{1},t_{2}]$ and $0<t_{2}-t_{1}\leq\displaystyle\frac{1}{k}$ . So, by construction,

E_{0}\ =\ \bigcup\limits_{k=1}^{\infty}E_{k}

and matters reduce to establishing that $\forall\ k,\lambda(E_{k})=0$ . To this end, let $\varepsilon>0$ and choose $P\in\mathcal{P}[a,b]:$

\sum\limits_{i=1}^{n}\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\ >\ \text{T}_{\underline{f}}[a,b]-\varepsilon.

Expanding $P$ if necessary, it an be assumed without loss of generality that

0\ <\ x_{i}-x_{i-1}\ \leq\ \frac{1}{k}\qquad(i=1,\ldots,n).

For each i, either $[x_{i-1},x_{i}]\cap E_{k}\neq\emptyset$ and then

s(x_{i})-s(x_{i-1})\ \geq\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert+\frac{x_{i}-x_{i-1}}{k},

or $[x_{i-1},x_{i}]\cap E_{k}=\emptyset$ and then

s(x_{i})-s(x_{i-1})\ \geq\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert.

Consequently

	$\displaystyle\text{T}_{\underline{f}}[a,b]\$	$\displaystyle=\ s(b)$
		$\displaystyle=\ s(x_{n})$
		$\displaystyle=\ \sum\limits_{i=1}^{n}(s(x_{i})-s(x_{i-1}))\qquad(s(x_{0})=s(a)=0)$
		$\displaystyle\geq\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert+\frac{1}{k}\lambda^{*}(E_{k})$
		$\displaystyle\geq\ \text{T}_{\underline{f}}[a,b]-\varepsilon+\frac{1}{k}\lambda^{*}(E_{k})$

$\implies$

\lambda^{*}(E_{k})\ \leq\ k\varepsilon\quad\implies\quad\lambda(E_{k})\ =\ 0\quad(\varepsilon\downarrow 0).

myn11

THEOREM Suppose that $C$ is continuous and rectifiable. Assume: $M>1$ $-$ then the $M$ -dimensional Lebesgue measure of $[C]$ is equal to 0.

myn12

NOTATION Let

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}

be a continuous curve. Given $\underline{x}\in[C]$ , let $N(\underline{f};\underline{x})$ be the number of points $x\in[a,b]$ (finite or infinite) such that $f(x)=\underline{x}$ and let $N(\underline{f};-)=0$ in the complement $\mathbb{R}^{M}-[C]$ of $[C]$ .

myn13

THEOREM

\ell(C)\ =\ \int\limits_{\mathbb{R}^{M}}\ N(\underline{f};-)\ \text{d}H^{1}.

[Note: $H^{1}$ is the $1$ -dimensional Hausdorff outer measure in $\mathbb{R}^{M}$ and

H^{1}([C])\ =\ \int\limits_{\mathbb{R}^{M}}\chi_{[C]}\ \text{d}H^{1}\ \leq\ \int\limits_{\mathbb{R}^{M}}N(\underline{f};-)\ \text{d}H^{1},

i.e.,

H^{1}([C])\ \leq\ \ell(C)

and it can happen that

H^{1}([C])\ <\ \ell(C).]

myn14

N.B. If $\underline{f}$ is one-to-one, then

N(\underline{f};-)\ =\ \chi_{[C]}

and when this is so,

H^{1}([C])=\ell(C).

$\boldsymbol{\S}$ 3. EQUIVALENCES

In what follows, by interval we shall understand a finite closed interval $\subset\mathbb{R}$ .

[Note: If $I,J$ are intervals and if $\partial I=\{a,b\}$ , $\partial J=\{c,d\}$ , then the agreement is that a homeomorphism $\phi:I\rightarrow J$ is sense preserving, i.e., sends $a$ to $c$ and $b$ to $d$ .]

myn1

DEFINITION Suppose given intervals $I$ , $J$ and curves $\underline{f}:I\rightarrow\mathbb{R}^{M}$ , $\underline{g}:J\rightarrow\mathbb{R}^{M}$ $-$ then $\underline{f}$ and $\underline{g}$ are said to be Lebesgue equivalent if there exists a homeomorphism $\phi:I\rightarrow J$ such that $\underline{f}=\underline{g}\circ\phi$ .

myn2

LEMMA If

\begin{cases}\ \underline{f}:[a,b]\rightarrow\mathbb{R}^{M}\\[4.0pt] \ \underline{g}:[c,d]\rightarrow\mathbb{R}^{M}\end{cases}

are Lebesgue equivalent and if

\begin{cases}\ C\longleftrightarrow\underline{f}\\[4.0pt] \ D\longleftrightarrow\underline{g}\end{cases},

then

\ell(C)\ =\ \ell(D).

PROOF The homeomorphism $\phi:[a,b]\rightarrow[c,d]$ induces a bijection

\begin{cases}\ \mathcal{P}[a,b]&\rightarrow\mathcal{P}[c,d]\\ \ \ P&\rightarrow Q\end{cases}.

Therefore

	$\displaystyle\ell(C)$	$\displaystyle=\sup\limits_{P\in\mathcal{P}[a,b]}\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle=\sup\limits_{P\in\mathcal{P}[a,b]}\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{g}(\phi(x_{i}))-\underline{g}(\phi(x_{i-1}))\right\rVert$
		$\displaystyle=\sup\limits_{Q\in\mathcal{P}[c,d]}\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{g}(y_{i})-\underline{g}(y_{i-1})\right\rVert$
		$\displaystyle=\ell(D).$

myn3

DEFINITION Suppose given intervals $I$ , $J$ and curves $\underline{f}:I\rightarrow\mathbb{R}^{M}$ , $\underline{g}:J\rightarrow\mathbb{R}^{M}$ $-$ then $\underline{f}$ and $\underline{g}$ are said to be Fréchet equivalent if for every $\varepsilon>0$ there exists a homeomorphism $\phi:I\rightarrow J$ such that

\left\lVert\underline{f}(x)-\underline{g}(\phi(x))\right\rVert\ <\ \varepsilon\qquad(x\in I).

myn4

REMARK It is clear that two Lebesgue equivalent curves are Fréchet equivalent but two Fréchet equivalent curves need not be Lebesgue equivalent.

myn5

LEMMA If

\begin{cases}\ \underline{f}:[a,b]\rightarrow\mathbb{R}^{M}\\[4.0pt] \ \underline{g}:[c,d]\rightarrow\mathbb{R}^{M}\end{cases}

are Fréchet equivalent and if

\begin{cases}\ C\longleftrightarrow\underline{f}\\[4.0pt] \ D\longleftrightarrow\underline{g}\end{cases},

then

\ell(C)\ =\ \ell(D).

PROOF For each $n=1,2,\ldots$ , there is a homeomorphism $\phi_{n}[a,b]\rightarrow[c,d]$ such that $\forall\ x\in[a,b]$ ,

\left\lVert\underline{f}(x)-\underline{g}(\phi_{n}(x))\right\rVert\ <\ \frac{1}{n}.

Put $\underline{f}_{\hskip 1.42271ptn}=\underline{g}\circ\phi_{n}$ , hence $\underline{f}_{\hskip 1.42271ptn}$ is Lebesgue equivalent to $\underline{g}$ (viz. $\underline{g}\circ\phi_{n}=\underline{g}\circ\phi_{n}\ldots$ ), thus if

C_{n}\longleftrightarrow\underline{f}_{\hskip 1.42271ptn},\quad D\longleftrightarrow\underline{g},

then from the above

\ell(C_{n})\ =\ \ell(D).

But $\forall\ x\in[a,b]$ ,

\left\lVert\underline{f}(x)-\underline{f}_{\hskip 1.42271ptn}(x)\right\rVert\ <\ \frac{1}{n},

i.e., $\underline{f}_{\hskip 1.42271ptn}\rightarrow\underline{f}$ pointwise, so

	$\displaystyle\ell(C)\$	$\displaystyle\leq\ \liminf\limits_{n\rightarrow\infty}\ \ell(C_{n})$
		$\displaystyle\leq\ \liminf\limits_{n\rightarrow\infty}\ \ell(D)$
		$\displaystyle\leq\ \ell(D).$

Analogously

\ell(D)\ \leq\ \ell(C).

Therefore

\ell(C)\ =\ \ell(D).

$\boldsymbol{\S}$ 4. FRÉCHET DISTANCE

Let

\begin{cases}\ C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}\\[8.0pt] \ D\longleftrightarrow\underline{g}:[c,d]\rightarrow\mathbb{R}^{M}\end{cases}

be two continuous curves.

myn1

NOTATION $\mathcal{H}$ is the set of all homeomorphisms

\phi:[a,b]\rightarrow[c,d]\qquad(\phi(a)=c,\ \phi(b)=d).

Given $\phi\in\mathcal{H}$ , the expression

\left\lVert\underline{f}(x)-\underline{g}(\phi(x))\right\rVert\qquad(a\leq x\leq b)

has an absolute maximum $M(\underline{f},\underline{g};\phi)$ .

myn2

DEFINITION The Fréchet distance between $C$ and $D$ , denoted $\left\lVert C,D\right\rVert$ , is

\inf\limits_{\phi\in\mathcal{H}}M(\underline{f},\underline{g});\phi).

[Note: In other words, $\left\lVert C,D\right\rVert$ is the infimum of all numbers $\varepsilon\geq 0$ with the property that there exists a homeomorphism $\phi\in\mathcal{H}$ such that

\left\lVert\underline{f}(x)-\underline{g}(\phi(x))\right\rVert\ \leq\ \varepsilon

for all $x\in[a,b].]$

myn3

N.B. If $\left\lVert C,D\right\rVert<\varepsilon$ , then there exists a $\phi\in\mathcal{H}$ such that

M(\underline{f},\underline{g};\phi)\ <\ \varepsilon.

myn4

LEMMA Let $C$ , $D$ , $C_{0}$ be continuous curves $-$ then

(i) $\left\lVert C,D\right\rVert\ \geq\ 0;$

(ii) $\left\lVert C,D\right\rVert\ =\ \left\lVert D,C\right\rVert;$

(iii) $\left\lVert C,D\right\rVert\ \leq\ \left\lVert C,C_{0}\right\rVert+\left\lVert C_{0},D\right\rVert;$

(iv) $\left\lVert C,D\right\rVert\ =\ 0\ \text{iff $C$ and $D$ are Fr\'{e}chet equivalent.}$

Therefore Fréchet distance is a premetric on the set of continuous curves with values in $\mathbb{R}^{M}$ .

myn5

THEOREM Let

\begin{cases}\ C_{n}\longleftrightarrow\underline{f_{n}}:[a_{n},b_{n}]\rightarrow\mathbb{R}^{M}\qquad(n=1,2,\ldots)\\[11.0pt] \ C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}\end{cases}

be continuous curves. Assume:

\left\lVert C_{n},C\right\rVert\rightarrow 0\qquad(n\rightarrow\infty).

Then

\qquad\ell(C)\ \leq\ \liminf\limits_{n\rightarrow\infty}\ \ell(C_{n}).

PROOF For every $n$ , there is a homeomorphism

\phi_{n}:[a,b]\rightarrow[a_{n},b_{n}]\qquad(\phi_{n}(a)=a_{n},\ \phi_{n}(b)=b_{n})

such that for all $x\in[a,b]$ ,

\left\lVert\underline{f}(x)-\underline{f_{n}}(\phi_{n}(x))\right\rVert\ <\ \left\lVert C,C_{n}\right\rVert+\frac{1}{n}.

Let

D_{n}\longleftrightarrow\underline{f_{n}}\circ\phi_{n}:[a,b]\rightarrow\mathbb{R}^{M}.

Then pointwise

\underline{f_{n}}\circ\phi_{n}\rightarrow\underline{f}

$\implies$

\ell(C)\ \leq\ \liminf\limits_{n\rightarrow\infty}\ell(D_{n}).

But $\ell(D_{n})=\ell(C_{n})$ , hence

\ell(C)\ \leq\ \liminf\limits_{n\rightarrow\infty}\ell(C_{n}).

In the set of continuous curves, introduce an equivalence relation by stipulating that $C$ and $D$ are equivalent provided $C$ and $D$ are Fréchet equivalent. The resulting set $\mathcal{E}_{F}$ of equivalence classes is then a metric space: If

\begin{cases}\ \{C\}\in\mathcal{E}_{F}\\[8.0pt] \ \{D\}\in\mathcal{E}_{F}\end{cases},

then

\left\lVert\{C\},\{D\}\right\rVert\ =\ \left\lVert C,D\right\rVert.

myn6

N.B. If $C$ , $C^{\prime}$ are Fréchet equivalent and if $D$ , $D^{\prime}$ are Fréchet equivalent, then

	$\displaystyle\left\lVert C,D\right\rVert\$	$\displaystyle\leq\ \left\lVert C,C^{\prime}\right\rVert+\left\lVert C^{\prime},D\right\rVert$
		$\displaystyle\leq\ \left\lVert C^{\prime},D\right\rVert$
		$\displaystyle\leq\ \left\lVert C^{\prime},D^{\prime}\right\rVert+\left\lVert D^{\prime},D\right\rVert$
		$\displaystyle=\ \left\lVert C^{\prime},D^{\prime}\right\rVert$

and in reverse

\left\lVert C^{\prime},D^{\prime}\right\rVert\ \leq\ \left\lVert C,D\right\rVert.

\left\lVert C,D\right\rVert\ =\ \left\lVert C^{\prime},D^{\prime}\right\rVert.

$\boldsymbol{\S}$ 5. THE REPRESENTATION THEOREM

Assume:

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}

is a curve which is continuous and rectifiable.

myn1

THEOREM There exists a continuous curve

D\longleftrightarrow\underline{g}:[c,d]\rightarrow\mathbb{R}^{M}

with the property that

\ell(D)\ =\ \ell(C)\qquad(<+\infty)

and

\ell(D)\ =\ \int\limits_{c}^{d}\ \left\lVert g^{\prime}\right\rVert,

where $g_{1},\ldots,g_{M}$ are absolutely continuous and in addition $\underline{f}$ and $\underline{g}$ are Fréchet equivalent.

Take $\ell(C)>0$ and define $\underline{g}$ via the following procedure. In the first place, the domain $[c,d]$ of $\underline{g}$ is going to be the interval $[0,\ell(C)]$ . This said, note that $s(x)$ is contant in an interval $[\alpha,\beta]$ iff $\underline{f}(x)$ is constant there as well. Next, for each point $s_{0}$ $(0\leq s_{0}\leq\ell(C))$ there is a maximal interval $\alpha\leq x\leq\beta\quad(a\leq\alpha\leq\beta\leq b)$ with $s(x)=s_{0}$ . Definition: $\underline{g}(s_{0})=\underline{f}(x)\quad(\alpha\leq x\leq\beta)$ .

myn2

LEMMA

\begin{cases}\ \underline{g}(s_{0^{-}})=\underline{g}(s_{0})\qquad(0<s_{0}\leq\ell(C))\\[4.0pt] \ \underline{g}(s_{0^{+}})=\underline{g}(s_{0})\qquad(0\leq s_{0}<\ell(C))\end{cases}.

Therefore

\underline{g}:[c,d]\rightarrow\mathbb{R}^{M}

is a continuous curve.

myn3

SUBLEMMA Suppose that $\phi_{n}:[A,B]\rightarrow[C,D]$ $(n=1,2,\ldots)$ converges uniformly to $\phi:[A,B]\rightarrow[C,D]$ . Let $\Phi:[C,D]\rightarrow\mathbb{R}^{M}$ be a continuous function $-$ then $\Phi\circ\phi_{n}$ converges uniformly to $\Phi\circ\phi$ .

PROOF Since $\Phi$ is uniformly continuous, given $\varepsilon>0$ , $\exists\ \delta>0$ such that

\left|u-v\right|\ <\ \delta\ \implies\ \left\lVert\Phi(u)-\Phi(v)\right\rVert\ <\ \varepsilon\qquad(u,v\in[C,D]).

Choose N:

n\ \geq\ N\ \implies\ \left|\phi_{n}(x)-\phi(x)\right|\ <\ \delta\qquad(x\in[A,B]).

Then

\left\lVert\Phi(\phi_{n}(x))-\Phi(\phi(x))\right\rVert\ <\ \varepsilon.

myn4

LEMMA $\underline{f}$ and $\underline{g}$ are Fréchet equivalent.

PROOF Approximate $s$ by quasilinear, strictly increasing functions $s_{n}(x)$ $(a\leq x\leq b)$ with $s_{n}(a)=0$ , $s_{n}(b)=\ell(C)$ and

\left|s_{n}(x)-s(x)\right|\ <\ \frac{1}{n}\qquad(n=1,2,\ldots).

Then

s_{n}:[a,b]\rightarrow[0,\ell(C)]

converges uniformly to

s:[a,b]\rightarrow[0,\ell(C)]

and

\underline{g}:[0,\ell(C)]\rightarrow\mathbb{R}^{M}

is continuous, so

\underline{g}\circ s_{n}\rightarrow\underline{g}\circ s

uniformly in $[a,b]$ , thus $\forall\ \varepsilon>0$ , $\exists\ N:n\geq N$

\implies\quad\left\lVert\underline{g}(s_{n}(x))-\underline{g}(s(x))\right\rVert\ <\ \varepsilon\qquad(a\leq x\leq b)

or still,

\left\lVert\underline{f}(x)-\underline{g}(s_{n}(x))\right\rVert\ <\ \varepsilon\qquad(a\leq x\leq b).

Since the $s_{n}$ are homeomorphisms, it follows that $\underline{f}$ and $\underline{g}$ are Fréchet equivalent.

myn5

LEMMA

0\leq u\ <\ v\ \leq\ \ell(C)

$\implies$

\left\lVert\underline{g}(v)-\underline{g}(u)\right\rVert\ =\ v-u

$\implies$

\left|g_{m}(v)-g_{m}(u)\right|\ \leq\ v-u\qquad(1\leq m\leq M).

Consequently $g_{1},\ldots,g_{M}$ are absolutely continuous (in fact, Lipschitz).

myn6

LEMMA

\ell(C)\ =\ \ell(D)\ =\ \int\limits_{0}^{\ell(D)}\ \left\lVert\underline{g}^{\prime}\right\rVert,

where $\left\lVert\underline{g}^{\prime}\right\rVert\leq 1$ .

	$\displaystyle 0\$	$\displaystyle=\ \ell(D)-\int\limits_{0}^{\ell(D)}\ \left\lVert\underline{g}^{\prime}\right\rVert$
		$\displaystyle=\ \int\limits_{0}^{\ell(D)}\ 1-\int\limits_{0}^{\ell(D)}\ \left\lVert\underline{g}^{\prime}\right\rVert$
		$\displaystyle=\ \int\limits_{0}^{\ell(D)}\ (1-\left\lVert\underline{g}^{\prime}\right\rVert)$

implying thereby that $\left\lVert\underline{g}^{\prime}\right\rVert=1$ almost everywhere.

$\boldsymbol{\S}$ 6. INDUCED MEASURES

myn1

NOTATION $\text{BO}[a,b]$ is the set of Borel subsets of $[a,b]$ .

Let

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}

be a curve, continuous and rectifiable.

myn2

LEMMA The interval function defined by the rule

[c,d]\rightarrow s(d)-s(c)\qquad([c,d]\subset[a,b])

can be extended to a measure $\mu_{C}$ on $\text{BO}[a,b]$ .

myn3

LEMMA For $m=1,\ldots,M$ , the interval function defined by the rule

[c,d]\rightarrow T_{f_{m}}[c,d]\qquad([c,d]\subset[a,b])

can be extended to a measure $\mu_{m}$ on $\text{BO}[a,b]$ .

myn4

FACT Given $S\in\text{BO}[a,b]$ ,

\mu_{m}(S)\leq\mu_{C}(S)\leq\mu_{1}(S)+\ldots+\mu_{M}(S).

myn5

LEMMA For $m=1,\ldots,M$ , the interval functions defined by the rule

\begin{cases}\ [c,d]\rightarrow T_{f_{m}}^{+}[c,d]\\[8.0pt] \ [c,d]\rightarrow T_{f_{m}}^{-}[c,d]\end{cases}\qquad([c,d]\subset[a,b])

can be extended to measures $\begin{cases}\ \mu_{m}^{+}\\[4.0pt] \ \mu_{m}^{-}\end{cases}$ on $\text{BO}[a,b]$ .

myn6

NOTATION Put

\nu_{m}\ =\ \mu_{m}^{+}-\mu_{m}^{-}\qquad(m=1,\ldots,M).

[Thus $\nu_{m}$ is a countably additive, totally finite set function on $\text{BO}[a,b].]$

myn7

RECOVERY PRINCIPLE For any $S\in\text{BO}[a,b]$ ,

\mu_{C}(S)\ =\ \sup\limits_{\{P\}}\ \sum\limits_{E\in P}\ \bigg{\{}\sum\limits_{m=1}^{M}\ \nu_{m}(E)^{2}\bigg{\}}^{\frac{1}{2}},

where the supremum is taken over all partitions $P$ of $S$ into disjoint Borel measurable sets $E$ .

myn8

FACT The set functions $\mu_{m},\ \mu_{m}^{+},\ \mu_{m}^{-},\ \nu_{m}$ are absolutely continuous w.r.t. $\mu_{C}$ .

myn9

NOTATION The corresponding Radon-Nikodym derivatives are denoted by

\beta_{m}=\frac{\text{d}\mu_{m}}{\text{d}\mu_{C}},\quad\begin{cases}\ \displaystyle\beta^{+}_{m}=\frac{\text{d}\mu_{m}^{+}}{\text{d}\mu_{C}}\\[15.0pt] \ \displaystyle\beta^{-}_{m}=\frac{\text{d}\mu_{m}^{-}}{\text{d}\mu_{C}}\end{cases},\ \displaystyle\theta_{m}=\frac{\text{d}\nu_{m}}{\text{d}\mu_{C}}.

myn10

CONVENTION The term almost everywhere (or measure 0) will refer to the measure space

([a,b],\text{BO}[a,b],\mu_{C}).

myn11

FACT

\begin{cases}\ \beta_{m}=\beta_{m}^{+}+\beta_{m}^{-}\\[8.0pt] \ \theta_{m}=\beta_{m}^{+}-\beta_{m}^{-}\end{cases}\qquad(m=1,\ldots,M)

almost everywhere.

myn12

NOTATION Let

\underline{\theta}=(\theta_{1},\ldots,\theta_{M}).

[Note: By definition,

\left\lVert\underline{\theta}(x)\right\rVert=(\theta_{1}(x)^{2}+\cdots+\theta_{M}(x)^{2})^{\frac{1}{2}}.]

myn13

NOTATION Given a linear orthogonal transformation $\lambda:\mathbb{R}^{M}\rightarrow\mathbb{R}^{M}$ , let $\overline{C}=\lambda\hskip 0.85355ptC$ .

myn14

N.B.

\mu_{\overline{C}}=\mu_{C}.

myn15

LEMMA

(\overline{\nu}_{1},\ldots,\overline{\nu}_{M})=\lambda(\nu_{1},\ldots,\nu_{M}).

myn16

APPLICATION

(\overline{\theta}_{1},\ldots,\overline{\theta}_{M})=\lambda(\theta_{1},\ldots,\theta_{M})

almost everywhere.

[Differentiate the preceding relation w.r.t. $\mu_{\overline{C}}=\mu_{C}.]$

myn17

LEMMA

\left|\theta_{m}\right|\ \leq\ 1\qquad(m=1,\ldots,M)

almost everywhere, so

\left\lVert\underline{\theta}\right\rVert\ \leq\ M^{\frac{1}{2}}

almost everywhere.

myn18

THEOREM

\left\lVert\underline{\theta}\right\rVert\ =\ 1

almost everywhere.

PROOF Let $0<\delta<1$ and let

S\ =\ \{x:\left\lVert\underline{\theta}(x)\right\rVert<1-\delta\}.

Then

\mu_{C}(S)\ =\ \sup\limits_{\{P\}}\ \sum\limits_{E\in P}\ \bigg{\{}\sum\limits_{m=1}^{M}\ \nu_{m}(E)^{2}\bigg{\}}^{\frac{1}{2}}.

But

	$\displaystyle\nu_{m}(E)\$	$\displaystyle=\ \int\limits_{E}\ \frac{\text{d}\nu_{m}}{\text{d}\mu_{C}}\ \text{d}\mu_{C}$
		$\displaystyle=\ \int\limits_{E}\ \theta_{m}\ \text{d}\mu_{C}.$

Therefore

	$\displaystyle\bigg{\{}\sum\limits_{m=1}^{M}\ \nu_{m}(E)^{2}\bigg{\}}^{\frac{1}{2}}\$	$\displaystyle=\ \bigg{\{}\sum\limits_{m=1}^{M}\ \bigg{(}\int\limits_{E}\ \theta_{m}\ \text{d}\mu_{C}\bigg{)}^{2}\bigg{\}}^{\frac{1}{2}}$
		$\displaystyle\leq\ \int\limits_{E}\ \bigg{\{}\sum\limits_{m=1}^{M}\ \theta_{m}^{2}\bigg{\}}^{\frac{1}{2}}\ \text{d}\mu_{C}$
		$\displaystyle=\ \int\limits_{E}\left\lVert\underline{\theta}(x)\right\rVert\ \text{d}\mu_{C}$
		$\displaystyle\leq\ (1-\delta)\int\limits_{E}\ \text{d}\mu_{C}$
		$\displaystyle=\ (1-\delta)\hskip 0.85355pt\mu_{C}(E).$

Since

S=\coprod E,

it follows that

\sum\limits_{E\in P}\ \bigg{\{}\sum\limits_{m=1}^{M}\ \nu_{m}(E)^{2}\bigg{\}}^{\frac{1}{2}}\ \leq\ (1-\delta)\hskip 1.42271pt\mu_{C}(S).

Taking the supremum over the $P$ then implies that

\mu_{C}(S)\ \leq\ (1-\delta)\hskip 1.42271pt\mu_{C}(S),

thus $\mu_{C}(S)=0$ and $\left\lVert\underline{\theta}(x)\right\rVert\geq 1$ almost everywhere (let $\displaystyle\delta=\frac{1}{2},\frac{1}{3},\ldots$ ). To derive a contradiction, take $M\geq 2$ and suppose that $\left\lVert\underline{\theta}(x)\right\rVert\geq 1+\delta>1$ on some set $T$ such that $\mu_{C}(T)>0$ $-$ then for some vector

\underline{\xi}\ =\ (\xi_{1},\ldots,\xi_{M})\in\mathbb{R}^{M}\qquad\left(\left\lVert\underline{\xi}\right\rVert=1\right),

the set

T(\underline{\xi})\ =\ \{x\in T:\bigg{|}\bigg{|}\frac{\underline{\theta}(x)}{\left\lVert\underline{\theta}(x)\right\rVert}-\underline{\xi}\bigg{|}\bigg{|}<\frac{\delta}{M^{2}}

has measure $\mu_{C}(T(\underline{\xi}))>0$ $($ see below $)$ . Let

\underline{\lambda_{j}}\ =\ (\lambda_{j_{1}},\ldots,\lambda_{j_{M}})\qquad(j=2,\ldots,M)

be unit vectors such that

\lambda=\begin{bmatrix}\xi_{1},&\ldots,&\xi_{M}\\ \lambda_{21},&\ldots,&\lambda_{2M}\\ \vdots&\vdots&\vdots\\ \lambda_{M1},&\ldots,&\lambda_{MM}\\ \end{bmatrix}

is an orthogonal matrix. Viewing $\lambda$ as a linear orthogonal transformation, form as above $\overline{C}=\lambda C$ , hence

(\overline{\theta}_{1},\ldots,\overline{\theta}_{M})=\lambda(\theta_{1},\ldots,\theta_{M}).

On $T(\underline{\xi})$ ,

	$\displaystyle\left\|\overline{\theta}_{j}\right\|\$	$\displaystyle=\ \left\|\lambda_{j_{1}}\theta_{1}+\cdots+\lambda_{j_{M}}\theta_{M}\right\|$
		$\displaystyle\leq\ \left\lVert\underline{\theta}\right\rVert\frac{\delta}{M^{2}}$
		$\displaystyle\leq\ M^{\frac{1}{2}}\frac{\delta}{M^{2}}$
		$\displaystyle\leq\ M\frac{\delta}{M^{2}}$
		$\displaystyle=\ \frac{\delta}{M},$

while

\left\lVert\overline{\underline{\theta}}\right\rVert\ \leq\ \left|\overline{\theta}_{1}\right|+\ldots+\left|\overline{\theta}_{M}\right|

$\implies$

	$\displaystyle\left\|\overline{\theta}_{1}\right\|\$	$\displaystyle\geq\left\lVert\overline{\underline{\theta}}\right\rVert-\left\|\overline{\theta}_{2}\right\|-\cdots-\left\|\overline{\theta}_{M}\right\|$
		$\displaystyle\geq\ (1+\delta)-(M-1)\frac{\delta}{M}$
		$\displaystyle=\ 1+\frac{\delta}{M}.$

However

\left|\overline{\theta}_{1}\right|\ \leq\ 1,

so we have a contradiction.

myn19

N.B. Let $\{\underline{\xi_{n}}:n\in\mathbb{N}\}$ be a dense subset of the unit sphere in $\text{U}(M)$ in $\mathbb{R}^{M}$ $($ thus $\forall\ n$ , $\left\lVert\underline{\xi_{n}}\right\rVert=1)$ . Given a point $x\in T$ , pass to

\frac{\underline{\theta}(x)}{\left\lVert\underline{\theta}(x)\right\rVert}\in\text{U}(M).

Then there exists a $\underline{\xi_{n_{x}}}:$

\left\lVert\frac{\underline{\theta}(x)}{\left\lVert\underline{\theta}(x)\right\rVert}-\underline{\xi_{n_{x}}}\right\rVert\ <\ \frac{\delta}{M^{2}},

a point in the $\displaystyle\frac{\delta}{M^{2}}$ $-$ neighborhood of

\frac{\underline{\theta}(x)}{\left\lVert\underline{\theta}(x)\right\rVert}

in $\text{U}(M)$ . Therefore

T\ =\ \bigcup\limits_{n=1}^{\infty}\ T(\underline{\xi_{n}})

$\implies$

0\ <\ \mu_{C}(T)\ \leq\ \sum\limits_{n=1}^{\infty}\ \mu_{C}(T(\underline{\xi_{n}}))

$\implies$ $\exists\ n:$

\mu_{C}(T(\underline{\xi_{n}}))\ >\ 0.

$\boldsymbol{\S}$ 7. TWO THEOREMS

Let

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}

be a curve, continuous and rectifiable.

Let $P\in\mathcal{P}[a,b]$ , say

P:a=x_{0}<x_{1}<\cdots<x_{n}=b.

myn1

DEFINITION Let $i=1,\ldots,n$ and for $m=1,\ldots,M$ let

\eta_{m}(x;P)\ =\ \frac{f_{m}(x_{i})-f_{m}(x_{i-1})}{\mu_{C}([x_{i-1},x_{i}])},

where $x_{i-1}<x<x_{i}$ if $\mu_{C}([x_{i-1},x_{i}])\neq 0$ and let

\eta_{m}(x;P)=0,

where $x_{i-1}<x<x_{i}$ if $\mu_{C}([x_{i-1},x_{i}])=0$ .

myn2

NOTATION

\underline{\eta}(x;P)\ =\ (\eta_{1}(x;P),\ldots,\eta_{M}(x;P)).

myn3

THEOREM

\int\limits_{a}^{b}\ \left\lVert\underline{\theta}(x)-\underline{\eta}(x;P)\right\rVert^{2}\ \text{d}\mu_{C}\ \leq\ 2\hskip 1.42271pt\bigg{[}\ell(C)\hskip 1.42271pt-\hskip 1.42271pt\sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\bigg{]}.

Proof: Given $P\in\mathcal{P}[a,b]$ , let $\sum^{\prime}$ denote a sum over intervals $[x_{i-1},x_{i}]$ , where

$\left\lVert\underline{\eta}(x;P)\right\rVert^{2}\neq 0$ and let $\sum^{\prime\prime}$ denote a sum over what remains. Now compute:

	$\displaystyle\int\limits_{a}^{b}\ \big{\|}\big{\|}\underline{\theta}(x)$	$\displaystyle-\underline{\eta}(x;P)\big{\|}\big{\|}^{2}\ \text{d}\mu_{C}\$
		$\displaystyle=\ {\sum}^{\prime}\ \int\limits_{x_{i-1}}^{x_{i}}\ \left\lVert\underline{\theta}(x)-\underline{\eta}(x;P)\right\rVert^{2}\ \text{d}\mu_{C}\ +\ {\sum}^{\prime\prime}\ \int\limits_{x_{i-1}}^{x_{i}}\ \left\lVert\underline{\theta}(x)\right\rVert^{2}\ \text{d}\mu_{C}$
		$\displaystyle=\ {\sum}^{\prime}\ \int\limits_{x_{i-1}}^{x_{i}}\ [\left\lVert\underline{\theta}(x)\right\rVert^{2}+\left\lVert\underline{\eta}(x;P)\right\rVert^{2}-2\hskip 0.85355pt\underline{\theta}(x)\cdot\underline{\eta}(x;P)]\ \text{d}\mu_{C}$
		$\displaystyle\hskip 85.35826pt\ +\ {\sum}^{\prime\prime}\int\limits_{x_{i-1}}^{x_{i}}\left\lVert\underline{\theta}(x)\right\rVert^{2}\ \text{d}\mu_{C}$
		$\displaystyle=\ {\sum}^{\prime}\ \int\limits_{x_{i-1}}^{x_{i}}\ [1+\left\lVert\underline{\eta}(x;P)\right\rVert^{2}-2\hskip 0.85355pt\underline{\theta}(x)\cdot\underline{\eta}(x;P)]\ \text{d}\mu_{C}\ +\ {\sum}^{\prime\prime}\ \int\limits_{x_{i-1}}^{x_{i}}1\ \text{d}\mu_{C}$
		$\displaystyle=\ {\sum}^{\prime}\ [\mu_{C}([x_{i-1},x_{i}])\ +\ \left[\frac{\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert}{\mu_{C}([x_{i-1},x_{i}])}\right]^{2}\ \mu_{C}([x_{i-1},x_{i}])$
		$\displaystyle\hskip 85.35826pt-2\hskip 0.85355pt\frac{\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert^{2}}{\mu_{C}([x_{i-1},x_{i}])}\ +\ {\sum}^{\prime\prime}\ \mu_{C}([x_{i-1},x_{i}])$
		$\displaystyle\leq\ \ell(C)\ -\ {\sum}^{\prime}\ \frac{\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert^{2}}{\mu_{C}([x_{i-1},x_{i}])}$
		$\displaystyle\leq\ \ell(C)\ -\ {\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle\hskip 85.35826pt\ +\ {\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\left(1-\frac{\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert}{\mu_{C}([x_{i-1},x_{i}])}\right)$
		$\displaystyle\leq\ \ell(C)\ -\ {\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle\qquad\qquad\qquad\qquad+{\sum}^{\prime}\mu_{C}([x_{i-1},x_{i}])\left(1-\frac{\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert}{\mu_{C}([x_{i-1},x_{i}])}\right)$
		$\displaystyle\leq\ \ell(C)-{\sum}^{\prime}\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle\hskip 85.35826pt\ +\ {\sum}^{\prime}\left(\mu_{C}([x_{i-1},x_{i}])-\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\right)$
		$\displaystyle\leq\ \ell(C)-{\sum}^{\prime}\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle\hskip 85.35826pt\ +\ {\sum}^{\prime}\ \mu_{C}([x_{i-1},x_{i}])\ -\ {\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle=\ \ell(C)\ +\ {\sum}^{\prime}\ \mu_{C}([x_{i-1},x_{i}])-2\hskip 1.42271pt{\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle=\ \ell(C)\ +\ \ell(C)-2\hskip 1.42271pt{\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle=\ 2\hskip 1.42271pt\left[\ell(C)\hskip 1.42271pt-\hskip 1.42271pt{\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\right]$
		$\displaystyle=\ 2\hskip 1.42271pt\bigg{[}\ell(C)-\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\bigg{]}.$

myn4

N.B. By definition, $\mu_{C}([x_{i-1},x_{i}])$ is the length of the restriction of $C$ to $[x_{i-1},x_{i}]$ , i.e.,

\mu_{C}([x_{i-1},x_{i}])\ =\ s(x_{i})-s(x_{i-1}).

Moreover

\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\ \leq\ s(x_{i})-s(x_{i-1}).

So, if $\mu_{C}([x_{i-1},x_{i}])=0$ , then

\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\ =\ 0\quad\implies\quad\underline{f}(x_{i})=\underline{f}(x_{i-1})

$\implies$

	$\displaystyle{\sum}^{\prime}\left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert\$	$\displaystyle=\ {\sum}^{\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert+{\sum}^{\prime\prime}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle=\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert.$

Abbreviate

\text{L}^{2}([a,b],\text{BO}([a,b]),\mu_{C})

\text{L}^{2}(\mu_{C}).

myn5

APPLICATION In $\text{L}^{2}(\mu_{C})$ ,

\lim\limits_{\left\lVert P\right\rVert\rightarrow 0}\ \underline{\eta}(-;P)=\underline{\theta}.

myn6

SETUP

• $C_{0}\longleftrightarrow\underline{f_{0}}:[a,b]\rightarrow\mathbb{R}^{M}$

is a curve, continuous and rectifiable.

• $C_{k}\longleftrightarrow\underline{f_{k}}:[a,b]\rightarrow\mathbb{R}^{M}\qquad(k=1,2,\ldots)$

is a sequence of curves, continuous and rectifiable.

Assumption: $\underline{f_{k}}$ converges uniformly to $\underline{f_{0}}$ in $[a,b]$ and

\lim\limits_{k\rightarrow\infty}\ \ell(C_{k})\ =\ \ell(C_{0}).

myn7

THEOREM

\lim\limits_{\left\lVert Q\right\rVert\rightarrow 0}\ \bigvee\limits_{a}^{b}\ (\underline{f_{k}};Q)\ =\ \ell(C_{k})\qquad(Q\in\mathcal{P}[a,b])

uniformly in $k$ , i.e., $\forall\ \varepsilon>0,\ \exists\ \delta>0$ such that

\left\lVert Q\right\rVert\ <\ \delta\implies\bigg{|}\bigvee\limits_{a}^{b}\ (\underline{f_{k}};Q)-\ell(C_{k})\bigg{|}\ <\ \varepsilon

for all $k=1,2,\ldots$ , or still,

\left\lVert Q\right\rVert\ <\ \delta\implies\ell(C_{k})-\bigvee\limits_{a}^{b}\ (\underline{f_{k}};Q)\ <\ \varepsilon

for all $k=1,2,\ldots\hskip 1.42271pt.$

The proof will emerge in the lines to follow. Start the process by choosing $\delta_{0}>0$ such that

\ell(C_{0})-\bigvee\limits_{a}^{b}\ (\underline{f_{0}};P_{0})\ <\ \frac{\varepsilon}{4}

provided $\left\lVert P_{0}\right\rVert<\delta_{0}$ . Consider a $P\in\mathcal{P}[a,b]:$

a=x_{0}<x_{1}<\cdots<x_{n}=b

with $\left\lVert P\right\rVert<\delta_{0}$ . Choose $\rho>0$ such that

\left\lVert\underline{f_{k}}(c)-\underline{f_{k}}(d)\right\rVert\ <\ \frac{\varepsilon}{4n}\qquad([c,d]\subset[a,b])

for all $k=0,1,2,\ldots$ , so long as $\left|c-d\right|<\rho$ (equicontinuity). Take a partition $Q\in\mathcal{P}[a,b]:$

a=y_{0}<y_{1}<\cdots<y_{m}=b

subject to

\left\lVert Q\right\rVert\ <\ \gamma\equiv\min\limits_{i=1,\ldots,n}\ \bigg{\{}\rho,\frac{x_{i}-x_{i-1}}{2}\bigg{\}}\qquad(\implies\left\lVert Q\right\rVert\ <\ \delta_{0}).

Put

\sigma_{k}\ =\ \sup\limits_{a\leq x\leq b}\ \left\lVert\underline{f_{k}}(x)-\underline{f_{0}}(x)\right\rVert

and let $k_{0}$ be such that

k>k_{0}\quad\implies\quad\sigma_{k}<\frac{\varepsilon}{4n}\quad\text{and}\quad\left|\ell(C_{k})-\ell(C_{0})\right|<\frac{\varepsilon}{4}.

The preparations complete, to minimize technicalities we shall suppose that each $I_{j}=[y_{j-1},y_{j}]$ is contained in just one $I_{i}=[x_{i-1},x_{i}]$ and write ${\sum}^{(i)}$ for a sum over all such $I_{j}$ $-$ then

	$\displaystyle\bigvee\limits_{a}^{b}\ (\underline{f_{k}};Q)\$	$\displaystyle=\ \sum\limits_{j=1}^{m}\ v(\underline{f_{k}};I_{j})$
		$\displaystyle=\ \sum\limits_{j=1}^{m}\ \left\lVert\underline{f_{k}}(y_{j})-\underline{f_{k}}(y_{j-1})\right\rVert$
		$\displaystyle=\ \sum\limits_{i=1}^{n}\ {\sum}^{(i)}\left\lVert\underline{f_{k}}(y_{j})-\underline{f_{k}}(y_{j-1})\right\rVert$
		$\displaystyle\geq\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f_{k}}(x_{i})-\underline{f_{k}}(x_{i-1})\right\rVert.$

myn8

SUBLEMMA Let $\underline{A},\ \underline{B},\ \underline{C},\ \underline{D}\in\mathbb{R}^{M}$ $-$ then

\left\lVert\underline{C}-\underline{D}\right\rVert>\left\lVert\underline{A}-\underline{B}\right\rVert-\left\lVert\underline{A}-\underline{C}\right\rVert-\left\lVert\underline{B}-\underline{D}\right\rVert.

[In fact,

	$\displaystyle\left\lVert\underline{A}-\underline{B}\right\rVert\$	$\displaystyle=\ \left\lVert\underline{A}-\underline{C}+\underline{C}-\underline{D}+\underline{D}-\underline{B}\right\rVert$
		$\displaystyle\leq\ \left\lVert\underline{A}-\underline{C}\right\rVert+\left\lVert\underline{C}-\underline{D}\right\rVert+\left\lVert\underline{B}-\underline{D}\right\rVert\hskip 1.42271pt].$

Take

\begin{cases}\ \underline{C}=\underline{f_{k}}(x_{i})\\[4.0pt] \ \underline{D}=\underline{f_{k}}(x_{i-1})\end{cases}\qquad\begin{cases}\ \underline{A}=\underline{f_{0}}(x_{i})\\[4.0pt] \ \underline{B}=\underline{f_{0}}(x_{i-1})\end{cases}.

Then

	$\displaystyle\left\lVert\underline{f_{k}}(x_{i})-\underline{f_{k}}(x_{i-1})\right\rVert$
	$\displaystyle\hskip 28.45274pt\geq\ \left\lVert\underline{f_{0}}(x_{i})\ -\ \underline{f_{0}}(x_{i-1})\right\rVert\ -\ \left\lVert\underline{f_{0}}(x_{i})-\underline{f_{k}}(x_{i})\right\rVert\ -\ \left\lVert\underline{f_{0}}(x_{i-1})-\underline{f_{k}}(x_{i-1})\right\rVert,$

thus

	$\displaystyle\sum\limits_{i=1}^{n}\ \left\lVert\underline{f_{k}}(x_{i})-\underline{f_{k}}(x_{i-1})\right\rVert\$	$\displaystyle\geq\ \ell(C_{0})-\frac{\varepsilon}{4}-n\sigma_{k}-n\sigma_{k}$
		$\displaystyle\geq\ \ell(C_{0})-\frac{\varepsilon}{4}-\frac{\varepsilon}{4}-\frac{\varepsilon}{4}$
		$\displaystyle=\ \ell(C_{0})-\frac{3\varepsilon}{4}.$

But

	$\displaystyle k>k_{0}$	$\displaystyle\implies\left\|\ell(C_{k}-\ell(C_{0}\right\|\ <\ \frac{\varepsilon}{4}$
		$\displaystyle\implies\ell(C_{k})-\frac{\varepsilon}{4}\ <\ \ell(C_{0}).$

Therefore

	$\displaystyle\ell(C_{0})-\frac{3\varepsilon}{4}\$	$\displaystyle>\ \ell(C_{k})\hskip 1.42271pt-\hskip 1.42271pt\frac{\varepsilon}{4}\hskip 1.42271pt-\hskip 1.42271pt\frac{3\varepsilon}{4}$
		$\displaystyle=\ \ell(C_{k})\hskip 1.42271pt-\hskip 1.42271pt\varepsilon.$

Thus: $\forall\ k>k_{0}$ ,

\ell(C_{k})-\bigvee\limits_{a}^{b}\ (\underline{f_{k}};Q)\ <\ \varepsilon\qquad(\left\lVert Q\right\rVert<\gamma).

Finally, for $k\leq k_{0}$ , let $\gamma_{k}$ be chosen so as to ensure that

\ell(C_{k})\hskip 1.42271pt-\hskip 1.42271pt\bigvee\limits_{a}^{b}\ (\underline{f_{k}};Q)\ <\ \varepsilon

for all partitions $Q$ with $\left\lVert Q\right\rVert<\gamma_{k}$ . Put now

\delta=\min\limits_{1,\ldots,k_{0}}\{\gamma_{1},\ldots\gamma_{k_{0}},\gamma\}.

Then

\left\lVert Q\right\rVert\ <\ \delta\quad\implies\quad\ell(C_{k})-\bigvee\limits_{a}^{b}\ (\underline{f_{k}};Q)\ <\ \varepsilon

for all $k=1,2,\ldots$ .

Changing the notation (replace $Q$ by $P$ ), $\forall\ \varepsilon>0$ , $\exists\ \delta>0$ such that

\left\lVert P\right\rVert\ <\ \delta\quad\implies\quad\ell(C_{k})-\bigvee\limits_{a}^{b}\ (\underline{f_{k}};P)\ <\ \varepsilon

for all $k=1,2,\ldots$ . Consequently

	$\displaystyle\int\limits_{a}^{b}\ \left\lVert\underline{\theta_{k}}(x)-\underline{\eta_{k}}(x;P)\right\rVert^{2}\ \text{d}\mu_{C_{k}}\$	$\displaystyle\leq\ 2\hskip 1.42271pt\bigg{[}\ell(C_{k})\hskip 1.42271pt-\hskip 1.42271pt\sum\limits_{i=1}^{n}\ \left\lVert\underline{f_{k}}(x_{i})-\underline{f_{k}}(x_{i-1})\right\rVert\bigg{]}$
		$\displaystyle=\ 2\hskip 1.42271pt\bigg{[}\ell(C_{k})\hskip 1.42271pt-\hskip 1.42271pt\bigvee\limits_{a}^{b}\ (\underline{f_{k}};P)\bigg{]}$
		$\displaystyle<\ 2\hskip 1.42271pt\varepsilon.$

$\boldsymbol{\S}$ 8. LINE INTEGRALS

Let

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}

be a curve, continuous and rectifiable.

Suppose that

F:[C]\times\mathbb{R}^{M}\rightarrow\mathbb{R},

say

F(\underline{x},\underline{t})\quad(\underline{x}\in[C],\underline{t}\in\mathbb{R}^{M}).

myn1

DEFINITION $F$ is a parametric integrand if $F$ is continuous in $(\underline{x},\underline{t})$ and $\forall\ K\geq 0$ ,

F(\underline{x},K\underline{t})\ =\ KF(\underline{x},\underline{t}).

myn2

EXAMPLE Let

F(\underline{x},\underline{t})\ =\ (t_{1}^{2}+\ldots+t_{M}^{2})^{\frac{1}{2}}.

myn3

EXAMPLE $(M=2)$ Let

F(x_{1},x_{2},t_{1},t_{2})\ =\ x_{1}t_{2}-x_{2}t_{1}.

myn4

N.B. If $F$ is a parametric integrand, then $\forall\ \underline{x}$ ,

F(\underline{x},0)\ =\ 0.

myn5

RAPPEL

\left\lVert\underline{\theta}\right\rVert=1

almost everywhere.

myn6

LEMMA Suppose that $F$ is a parametric integrand $-$ then the integral

I(C)\ \equiv\ \int\limits_{a}^{b}F(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x))\ \text{d}\mu_{C}

exists.

PROOF $[C]\times U(M)$ is a compact set on which $F$ is bounded. Since

(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x))\in[C]\times U(M)

almost everywhere, the function

F(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x))

is Borel measurable and essentially bounded w.r.t the measure $\mu_{C}$ . Therefore

I(C)\equiv\int\limits_{a}^{b}F(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x))\ \text{d}\mu_{C}

exists.

[Note: The requirement ``homogeneous of degree 1'' in $t$ plays no role in the course of establishing the existence of $I(C)$ . It will, however, be decisive in the considerations to follow.]

Let $P\in\mathcal{P}[a,b]$ and let $\xi_{i}$ be a point in $[x_{i-1},x_{i}]$ $(i=1,\ldots,n)$ .

myn7

THEOREM If $F$ is a parametric integrand, then

\lim\limits_{\left\lVert P\right\rVert\rightarrow 0}\ \sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))

exists and equals $I(C)$ , denote it by the symbol

\int\limits_{C}F,

and call it the line integral of $F$ along $C$ .

PROOF Fix $\varepsilon>0$ and let $\text{B}(M)$ be the unit ball in $\mathbb{R}^{M}$ . Put

M_{F}=\sup\limits_{[C]\times\text{B}(M)}\left|F\right|.

Choose $\gamma>0:$

\begin{cases}\ \left\lVert\underline{x_{1}}-\underline{x_{2}}\right\rVert<\gamma\qquad&(\underline{x_{1}},\underline{x_{2}}\in[C])\\[8.0pt] \ \left\lVert\underline{t_{1}}-\underline{t_{2}}\right\rVert<\gamma\qquad&(\underline{t_{1}},\underline{t_{2}}\in\text{B}(M))\end{cases}

$\implies$

\left|F(\underline{x_{1}},\underline{t_{1}})-F(\underline{x_{2}},\underline{t_{2}})\right|\ <\ \frac{\varepsilon}{3\ell(C)}.

Introduce $\underline{\eta}(x;P)$ and set

g(x;P)=F(\underline{f}(\xi_{i}),\underline{\eta}(x;P))

if $x_{i-1}<x<x_{i}$ $-$ then

	$\displaystyle\int\limits_{a}^{b}g(x;P)\ \text{d}\mu_{C}\$	$\displaystyle=\ \sum\limits_{i=1}^{n}\ F\left(\underline{f}(\xi_{i}),\frac{\underline{f}(x_{i})-\underline{f}(x_{i-1})}{\mu_{C}([x_{i-1},x_{i}])}\right)\mu_{C}([x_{i-1},x_{i}])$
		$\displaystyle=\ \sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))$

modulo the usual convention if $\mu_{C}([x_{i-1},x_{i}])=0$ . Recall now that in $\text{L}^{2}(\mu_{C})$ ,

\lim\limits_{\left\lVert P\right\rVert\rightarrow 0}\underline{\eta}(-;P)=\underline{\theta},

hence $\underline{\eta}(-;P)$ converges in measure to $\underline{\theta}$ , so there is a $\rho>0$ such that for all $P$ with $\left\lVert P\right\rVert<\rho$ ,

\left\lVert\underline{\theta}(x)-\underline{\eta}(x;P)\right\rVert<\gamma

except on a set $S_{P}$ of measure

\mu_{C}(S_{P})\ <\ \frac{\varepsilon}{3M_{F}}.

Define $\sigma:$

\left|t_{1}-t_{2}\right|<\sigma\quad\implies\quad\left\lVert f(t_{1})-f(t_{2})\right\rVert<\gamma.

Let $\delta=\min(\sigma,\rho)$ and let $P$ be any partition with $\left\lVert P\right\rVert<\delta$ $-$ then

	$\displaystyle I(C)-\sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),$	$\displaystyle\underline{f}(x_{i})-\underline{f}(x_{i-1}))\$
		$\displaystyle=\ \int\limits_{a}^{b}\ F(\underline{f}(x),\underline{\theta}(x))\ \text{d}\mu_{C}-\int\limits_{a}^{b}\ g(x;P)\ \text{d}\mu_{C}$
		$\displaystyle=\ \int\limits_{a}^{b}\ [F(\underline{f}(x),\underline{\theta}(x))-g(x;P)]\ \text{d}\mu_{C}.$

By definition, $\delta\leq\rho$ , hence

\left\lVert\underline{\theta}(x)-\underline{\eta}(x;P)\right\rVert\ <\ \gamma

except in $S_{P}$ , and

\left\lVert\underline{f}(x)-\underline{f}(\xi_{i})\right\rVert\ <\ \gamma

since

\left|x-\xi_{i}\right|<\gamma\quad(x_{i-1}\leq x\leq x_{i}).

To complete the argument, take absolute values:

	$\displaystyle\bigg{\|}I(C)\hskip 1.42271pt-\hskip 1.42271pt\sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),$	$\displaystyle\hskip 1.42271pt\underline{f}(x_{i})-\underline{f}(x_{i-1}))\bigg{\|}$
		$\displaystyle\leq\ \int\limits_{a}^{b}\ \left\|F(\underline{f}(x),\underline{\theta}(x))-g(x;P)\right\|\ \text{d}\mu_{C}$
		$\displaystyle=\ \int\limits_{[a,b]-S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}+\int\limits_{S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}.$

• On $[a,b]$ $-$ $S_{P}$ at an index $i$ ,

	$\displaystyle\left\|F(\underline{f}(x),\underline{\theta}(x))-g(x;P)\right\|\$	$\displaystyle=\ \left\|F(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x))-F(\underline{f}(\xi_{i}),\hskip 0.85355pt\underline{\eta}(x;P))\right\|$
		$\displaystyle\leq\ \frac{\varepsilon}{3\hskip 0.85355pt\ell(C)}.$

Here, of course, up to a set of measure 0,

\underline{\theta}(x)\in\text{B}(M)\text{ and }\eta(x;P)\in\text{B}(M).

Therefore

	$\displaystyle\int\limits_{[a,b]-S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}\$	$\displaystyle\leq\ \frac{\varepsilon}{3\hskip 0.85355pt\ell(C)}$
		$\displaystyle=\ \frac{\varepsilon}{3}.$

• On $S_{P}$ ,

\begin{cases}\ \left|F(\underline{f}(x),\underline{\theta}(x))\right|\hskip 1.42271pt\leq\hskip 1.42271ptM_{F}\\[8.0pt] \ \left|F(\underline{f}(\xi_{i}),\underline{\eta}(x;P))\right|\hskip 1.42271pt\leq\hskip 1.42271ptM_{F}\end{cases}.\\

Therefore

	$\displaystyle\int\limits_{S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}\$	$\displaystyle\leq\ 2M_{F}\int\limits_{S_{P}}\ 1\ \text{d}\mu_{C}$
		$\displaystyle=\ 2\hskip 1.42271ptM_{F}\hskip 1.42271pt\mu_{C}(S_{P})$
		$\displaystyle<\ 2\hskip 1.42271ptM_{F}\hskip 1.42271pt\frac{\varepsilon}{3M_{F}}$
		$\displaystyle=\ \frac{2\varepsilon}{3}.$

So in conclusion,

	$\displaystyle\int\limits_{[a,b]-S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}\hskip 1.42271pt+\hskip 1.42271pt\int\limits_{S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}\$	$\displaystyle<\ \frac{\varepsilon}{3}+\frac{2\varepsilon}{3}$
		$\displaystyle=\ \varepsilon\qquad(\left\lVert P\right\rVert<\delta)$

and

I(C)\ =\ \int\limits_{C}F.

myn8

N.B. The end result is independent of the choice of the $\xi_{i}$ .

myn9

THEOREM If $f_{1},\ldots,f_{M}\in\text{AC}[a,b]$ , then for any parametric integrand $F$ ,

\int\limits_{C}F\ =\ \int\limits_{a}^{b}F(f_{1}(x),\ldots,f_{M}(x),f_{1}^{\prime}(x),\ldots,f_{M}^{\prime}(x))\ \text{d}x,

the integral on the right being in the sense of Lebesgue.

PROOF The absolute continuity of the $f_{m}$ implies that

\mu_{C}([c,d])\ =\ \int\limits_{c}^{d}\left\lVert f^{\prime}\right\rVert\ \text{d}x

for every subinterval $[c,d]\subset[a,b]$ , hence $\mu_{C}$ is absolutely continuous w.r.t Lebesgue measure. It is also true that $\nu_{m}$ is absolutely continuous w.r.t Lebesgue measure. This said, write

f_{m}^{\prime}\ =\ \frac{\text{d}f_{m}}{\text{d}x}\ =\ \frac{\text{d}\nu_{m}}{\text{d}x}\ =\ \frac{\text{d}\nu_{m}}{\text{d}\mu_{C}}\frac{\text{d}\mu_{C}}{\text{d}x}\ =\ \theta_{m}\frac{\text{d}\mu_{C}}{\text{d}x}.

Then

	$\displaystyle I(C)$	$\displaystyle=\ \int\limits_{a}^{b}\ F(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x))\ \text{d}\mu_{C}$
		$\displaystyle=\ \int\limits_{a}^{b}\ F(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x))\frac{\text{d}\mu_{C}}{\text{d}x}\text{d}x$
		$\displaystyle=\ \int\limits_{a}^{b}\ F\left(\underline{f}(x),\hskip 0.85355pt\underline{\theta}(x)\frac{\text{d}\mu_{C}}{\text{d}x}\right)\ \text{d}x,$

where

\frac{\text{d}\mu_{C}}{\text{d}x}\ =\ \left\lVert\underline{f}^{\prime}\right\rVert\ \geq\ 0.

Continuing

	$\displaystyle I(C)\$	$\displaystyle=\ \int\limits_{a}^{b}\ F\left(f_{1}(x),\ldots,f_{M}(x),\hskip 0.85355pt\theta_{1}(x)\frac{\text{d}\mu_{C}}{\text{d}x},\ldots,\theta_{M}(x)\frac{\text{d}\mu_{C}}{\text{d}x}\right)\text{d}x$
		$\displaystyle=\ \int\limits_{a}^{b}\ F(f_{1}(x),\ldots,f_{M}(x),f_{1}^{\prime}(x),\ldots,f_{M}^{\prime}(x))\ \text{d}x,$

the integrals being in the sense of Lebesgue.

Let

\begin{cases}\ C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}\\ \ D\longleftrightarrow\underline{g}:[a,b]\rightarrow\mathbb{R}\end{cases}

be curves, continuous and rectifiable.

myn10

RAPPEL If $C$ and $D$ are Fréchet equivalent, then

[C]\ =\ [D]\quad\text{and}\quad\ell(C)\ =\ \ell(D).

myn11

THEOREM If $C$ and $D$ are Fréchet equivalent and if $F$ is a parametric integrand, then

\int\limits_{C}F=\int\limits_{D}F.

PROOF Fix $\varepsilon>0$ and choose $\delta>0:$

• $P\in\mathcal{P}[a,b]$ & $\left\lVert P\right\rVert<\delta$ $\implies$

\bigg{|}I(C)-\sum\limits_{i=1}^{n}F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))\bigg{|}\ <\ \frac{\varepsilon}{3}.

• $Q\in\mathcal{P}[c,d]$ & $\left\lVert Q\right\rVert<\delta$ $\implies$

\bigg{|}I(D)-\sum\limits_{j=1}^{m}F(\underline{f}(\xi_{j}),\underline{f}(y_{j})-\underline{f}(y_{j-1}))\bigg{|}\ <\ \frac{\varepsilon}{3}.

Fix $P$ and $Q$ satisfying these conditions and let $k$ be the number of intervals in $P$ and let $\ell$ be the number of intervals in $Q$ . Fix $\gamma>0$ such that

\left|F(\underline{x}_{1},\underline{t}_{1})-F(\underline{x}_{2},\underline{t}_{2})\right|\ <\ \frac{\varepsilon}{3(k+\ell)}

when

\left\lVert\underline{x}_{1}-\underline{x}_{2}\right\rVert\ <\ \gamma\qquad(\underline{x}_{1},\underline{x}_{2}\in[C]=[D])

and

\left\lVert\underline{t}_{1}-\underline{t}_{2}\right\rVert\ <\ 2\gamma\qquad(\left\lVert\underline{t}_{1}\right\rVert\ \leq\ \ell(C),\ \left\lVert\underline{t}_{2}\right\rVert\leq\ell(D)).

Let $\phi:[a,b]\rightarrow[c,d]$ be a homeomorphism $(\phi(a)=c,\ \phi(b)=d)$ such that

\left\lVert\underline{f}(x)-\underline{g}(\phi(x))\right\rVert\ <\ \gamma\qquad(x\in[a,b]).

Let

P^{*}:\ a=x_{0}^{*}<x_{1}^{*}<\cdots<x_{r}^{*}=b

be the partition obtained from $P$ by adjoining the images under $\phi^{-1}$ of the partition points of $Q$ . Let

Q^{*}:\ c=y_{0}^{*}<y_{1}^{*}<\cdots<y_{s}^{*}=d

be the partition obtained from $Q$ by adjoining the images under $\phi$ of the partition points of $P$ . So, by construction, $r=s$ , either one is $\leq k+\ell$ , and $y_{p}^{*}=\phi(x_{p}^{*})\ (p=0,1,\ldots,q)$ . Choose a point $\xi_{p}\in[x_{p-1}^{*},x_{p}^{*}]$ and work with

\underline{f}(\xi_{p})\quad\text{and}\quad\underline{g}(\phi(\xi_{p})).

Then

	$\displaystyle\left\|I(C)-I(D)\right\|\$	$\displaystyle\leq\ \bigg{\|}I(C)-\sum\limits_{p=1}^{q}\ F(\underline{f}(\xi_{p}),\underline{f}(x_{p}^{})-\underline{f}(x_{p-1}^{}))\bigg{\|}$
		$\displaystyle\qquad+\sum\limits_{p=1}^{q}\ \left\|F(\underline{f}(\xi_{p}),\underline{f}(x_{p}^{})-\underline{f}(x_{p-1}^{}))-F(\underline{g}(\phi(\xi_{p})),\underline{g}(y_{p}^{})-\underline{g}(y_{p-1}^{}))\right\|$
		$\displaystyle\qquad\qquad+\bigg{\|}\sum\limits_{p=1}^{q}\ F(\underline{g}(\phi(\xi_{p})),\underline{g}(y_{p}^{})-\underline{g}(y_{p-1}^{}))-I(D)\bigg{\|}.$

Since

\begin{cases}\ \left\lVert P^{*}\right\rVert\ \leq\ \left\lVert P\right\rVert\ <\ \delta\\[8.0pt] \ \left\lVert Q^{*}\right\rVert\ \leq\ \left\lVert Q\right\rVert\ <\ \delta\end{cases},

the first and third terms are each $\displaystyle<\frac{\varepsilon}{3}$ . As for the middle term,

\left\lVert\underline{f}(\xi_{p})-\underline{g}(\phi(\xi_{p}))\right\rVert<\gamma

and

	$\displaystyle\left\lVert\underline{f}(x_{p}^{})-\underline{f}(x_{p-1}^{})-\underline{g}(y_{p}^{})+\underline{g}(y_{p-1}^{})\right\rVert\$
	$\displaystyle\hskip 56.9055pt\leq\ \left\lVert\underline{f}(x_{p}^{})-\underline{g}(y_{p}^{})\right\rVert+\left\lVert\underline{f}(x_{p-1}^{})-\underline{g}(y_{p-1}^{})\right\rVert$
	$\displaystyle\hskip 56.9055pt=\ \left\lVert\underline{f}(x_{p}^{})-\underline{g}(\phi(x_{p}^{}))\right\rVert+\left\lVert\underline{f}(x_{p-1}^{})-\underline{g}(\phi(x_{p-1}^{}))\right\rVert$
	$\displaystyle\hskip 56.9055pt<\ \gamma+\gamma$
	$\displaystyle\hskip 56.9055pt=\ 2\gamma.$

Therefore the middle term is

<\ q\hskip 1.42271pt\frac{\varepsilon}{3(k+\ell)}\ =\ \frac{q}{k+\ell}\ \frac{\varepsilon}{3}\ <\ \frac{\varepsilon}{3}.

And finally

\left|I(C)-I(D)\right|\ <\ \frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}\ =\ \varepsilon

$\implies$

I(C)\ =\ I(D)\qquad(\varepsilon\downarrow 0)

$\implies$

\int\limits_{C}\ F\ =\ \int\limits_{D}\ F.

myn12

SETUP

• $C_{0}\longleftrightarrow\underline{f_{0}}:[a,b]\rightarrow\mathbb{R}^{M}$

is a curve, continuous and rectifiable.

• $C_{k}\longleftrightarrow\underline{f_{k}}:[a,b]\rightarrow\mathbb{R}^{M}\qquad(k=1,2,\ldots)$

is a sequence of curves, continuous and rectifiable.

Assumption: $\underline{f_{k}}$ converges uniformly to $\underline{f_{0}}$ in $[a,b]$ and

\lim\limits_{k\rightarrow\infty}\ell(C_{k})=\ \ell(C_{0}).

myn13

THEOREM

\lim\limits_{k\rightarrow\infty}\ I(C_{k})\ =\ I(C_{0})

or still,

\lim\limits_{k\rightarrow\infty}\ \int\limits_{C_{k}}\ F\ =\ \int\limits_{C_{0}}\ F.

$\boldsymbol{\S}$ 9. QUASI ADDITIVITY

myn1

DATA $A$ is a nonempty set, $\mathcal{I}=\{I\}$ is a nonempty collection of subsets of $A$ , $\mathcal{D}=\{D\}$ is a nonempty collection of nonempty finite collections of $D=[I]$ of sets $I\in\mathcal{I}$ , and $\delta$ is a real valued function defined on $\mathcal{D}$ .

myn2

DEFINITIONS The sets $I\in\mathcal{I}$ are called intervals, the collections $D\in\mathcal{D}$ are called systems, and the function $\delta$ is called a mesh.

myn3

ASSUMPTIONS $A$ is a nonempty topological space, each interval $I$ has a nonempty interior, the intervals of each system $D$ are nonoverlapping $:$ $I_{1},I_{2}\in D$ , $I_{1}\neq I_{2}$

\implies\qquad\begin{cases}\ \text{int}\ I_{1}\cap\ c\ell I_{2}\ =\ \emptyset\\[4.0pt] \ c\ell\ I_{1}\cap\text{int}\ I_{2}\ =\ \emptyset\end{cases}.

myn4

ASSUMPTION For each system $D$ , $0<\delta(D)<+\infty$ , and each $\varepsilon>0$ , there are systems with $\delta(D)<\varepsilon$ .

myn5

REMARK In the presence of $\delta$ , one is able to convert $\mathcal{D}$ into a directed set with direction `` $\gg$ '' by defining $D_{2}\gg D_{1}$ iff $\delta(D_{2})<\delta(D_{1})$ .

myn6

EXAMPLE Take $A=[a,b]$ and let $\mathcal{I}=\{I\}$ be the collection of all closed subintervals of $[a,b]$ . Take for $\mathcal{D}$ the class of all partitions $D$ of $[a,b]$ , i.e., $\mathcal{D}=\mathcal{P}[a,b]$ , and let $\delta(D)$ be the norm of $D$ .

[Note: Strictly speaking, an element of $\mathcal{P}[a,b]$ is a finite set $P=\{x_{0},\ldots,x_{n}\}$ , where

a=x_{0}<x_{1}<\cdots<x_{n}=b,

the associated element $D$ in $\mathcal{D}$ being the set

[x_{i-1},x_{i}]\qquad(i=1,\ldots,n).]

myn7

DEFINITION An interval function is a function $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ .

[Note: Associated with $\phi$ is the interval function $\left\lVert\phi\right\rVert$ , as well as the

\phi_{m},\ \left|\phi_{m}\right|,\ \begin{cases}\ \phi_{m}^{+}\\[4.0pt] \ \phi_{m}^{-}\end{cases}\qquad(m=1,\ldots,M).]

myn8

NOTATION Given an interval function $\phi$ , a subset $S\subset A$ , and a system $D=[I]$ , put

\sum\ [\phi,S,D]\ =\ \sum\limits_{I}s(I,S)\ \phi(I),

where $\displaystyle\sum\limits_{I}$ ranges over all $I\in D$ and $s(I,S)=1$ or 0 depending on whether $I\subset S$ or $I\not\subset S$ .

[Note: Take for $S$ the empty set $\emptyset$ $-$ then $I\subset\emptyset$ is inadmissible ( $I$ has nonempty interior) and $I\not\subset S$ gives rise to zero. Therefore

\sum\ [\phi,\emptyset,D]\ =\ 0.]

myn9

N.B. The absolute situation is when $S=A$ , thus in this case,

\sum\ [\phi,A,D]\ \equiv\ \sum\ [\phi,D]\ =\ \sum\limits_{I}\ \phi(I).

myn10

DEFINITION Given an interval function $\phi$ and a subset $S\subset A$ , the BC-integral of $\phi$ over $S$ is

\lim\limits_{\delta(D)\rightarrow 0}\ \sum\ [\phi,S,D]

provided the limit exists in $\mathbb{R}^{M}$ .

[Note: B = Burkill and C = Cesari.]

myn11

NOTATION The BC-integral of $\phi$ over $S$ is denoted by

\text{BC}\int\limits_{S}\phi.

myn12

EXAMPLE

\text{BC}\int\limits_{\emptyset}\ \phi\ =\ \underline{0}\quad(\in\mathbb{R}^{M}).

myn13

DEFINITION An interval function $\phi$ is quasi additive on $S$ if for each $\varepsilon>0$ there exists $\eta(\varepsilon,S)>0$ such that if $D_{0}=[I_{0}]$ is any system subject to $\delta(D_{0})<\eta(\varepsilon,S)$ , there also exists $\lambda(\varepsilon,S,D_{0})>0$ such that for every system $D=[I]$ with $\delta(D)<\lambda(\varepsilon,S,D_{0})$ , the relations

	$\displaystyle(\text{qa}_{1}-S)\hskip 1.42271pt\sum\limits_{I_{0}}s(I_{0},S)\ \big{\|}\big{\|}\sum\limits_{I}s(I,I_{0}),\phi(I)-\phi(I_{0})\big{\|}\big{\|}\ <\ \varepsilon$
	$\displaystyle(\text{qa}_{2}-S)\hskip 1.42271pt\sum\limits_{I}s(I,S)\ \bigg{[}1\hskip 1.42271pt-\hskip 1.42271pt\sum\limits_{I_{0}}s(I,I_{0}),s(I_{0},S)\bigg{]}\left\lVert\phi(I)\right\rVert\ <\ \varepsilon$

obtain.

myn14

N.B. In the absolute situation, matters read as follows: An interval function $\phi$ is quasi additive if for each $\varepsilon>0$ there exists $\eta(\varepsilon)>0$ such that if $D_{0}=[I_{0}]$ is any system subject to $\delta(D_{0})<\eta(\varepsilon)$ there exists $\lambda(\varepsilon,D_{0})>0$ such that for every system $D=[I]$ with $\delta(D)<\lambda(\varepsilon,D_{0})$ , the relations

	$\displaystyle(\text{qa}_{1}-A)\ \sum\limits_{I_{0}}\ \big{\|}\big{\|}\sum\limits_{I\subset I_{0}}\ \phi(I)-\phi(I_{0})\big{\|}\big{\|}\ <\ \varepsilon$
	$\displaystyle(\text{qa}_{2}-A)\ \sum\limits_{I\not\subset I_{0}}\ \left\lVert\phi(I)\right\rVert\ <\ \varepsilon$

obtain.

[Note: The sum

\sum\limits_{I\not\subset I_{0}}\left\lVert\phi(I)\right\rVert

is over all $I\in D$ , $I\not\subset I_{0}$ for any $I_{0}\in D_{0}.]$

So, under the preceding conditions,

\sum\limits_{I}\ \phi(I)-\sum\limits_{I_{0}}\ \phi(I_{0})\ =\ \sum\limits_{I_{0}}\ \bigg{[}\sum\limits_{I\subset I_{0}}\ \phi(I)-\phi(I_{0})\bigg{]}+\sum\limits_{I\not\subset I_{0}}\ \phi(I)

$\implies$

\bigg{|}\bigg{|}\sum\limits_{I}\ \phi(I)-\sum\limits_{I_{0}}\ \phi(I_{0})\bigg{|}\bigg{|}\ <\ 2\varepsilon.

myn15

THEOREM If $\phi$ is quasi additive on $S$ , then

\text{BC}\int\limits_{S}\phi

exists.

PROOF To simplify the combinatorics, take $S=A$ . Given $\varepsilon>0$ , let $\eta(\varepsilon)$ , $D_{0}$ , $\lambda(\varepsilon,D_{0})$ be per $\text{qa}_{1}$ - $A$ , $\text{qa}_{2}$ - $A$ and suppose that $D_{1},\ D_{2}\in\mathcal{D}$ , where

\begin{cases}\ \delta(D_{1})\ <\ \lambda(\varepsilon,D_{0})\\[4.0pt] \ \delta(D_{2})\ <\ \lambda(\varepsilon,D_{0})\end{cases}.

Then

\begin{cases}\displaystyle\ \big{|}\big{|}\sum\limits_{I_{1}}\phi(I_{1})-\sum\limits_{I_{0}}\phi(I_{0})\big{|}\big{|}\ <\ 2\hskip 0.85355pt\varepsilon\\[26.0pt] \displaystyle\ \big{|}\big{|}\sum\limits_{I_{2}}\phi(I_{2})-\sum\limits_{I_{0}}\phi(I_{0})\big{|}\big{|}\ <\ 2\hskip 0.85355pt\varepsilon\end{cases}

$\implies$

\big{|}\big{|}\sum\limits_{I_{1}}\phi(I_{1})-\sum\limits_{I_{2}}\phi(I_{2})\big{|}\big{|}\ <\ 4\hskip 0.85355pt\varepsilon.

Therefore $\displaystyle\text{BC}\int\limits_{A}\phi$ exists.

myn16

REMARK

• If the $\phi_{m}$ $(m=1,\ldots,M)$ are quasi additive, then $\phi$ is quasi additive.

• If the $\left|\phi_{m}\right|$ $(m=1,\ldots,M)$ are quasi additive, then $\left\lVert\phi\right\rVert$ is quasi additive.

myn17

DEFINITION $A$ real valued interval function $\psi$ is quasi subadditive on $S$ if for each $\varepsilon>0$ there exists $\eta(\varepsilon,S)>0$ such that if $D_{0}=[I_{0}]$ is any system subject to $\delta(D_{0})<\eta(\varepsilon,S)$ there also exists $\lambda(\varepsilon,S,D_{0})>0$ such that for every system $D=[I]$ with $\delta(D)<\lambda(\varepsilon,S,D_{0})$ the relation

(\text{qsa-}S)\ \sum\limits_{I_{0}}\ s(I_{0},S)\ \bigg{[}\sum\limits_{I}\ s(I,I_{0})\psi(I)-\psi(I_{0})\bigg{]}^{-}\ <\ \varepsilon

obtains.

myn18

N.B. In the absolute situation, matters read as follows $:$ …

(\text{qsa-}A)\ \sum\limits_{I_{0}}\ \bigg{[}\sum\limits_{I\subset I_{0}}\ \psi(I)-\psi(I_{0})\bigg{]}^{-}\ <\ \varepsilon.

myn19

LEMMA If $\psi:\mathcal{D}\rightarrow\mathbb{R}_{\geq 0}$ is nonnegative and quasi subadditive on $S$ , then

\text{BC}\int\limits_{S}\psi

exists $(+\infty$ is a permissible value $)$ .

myn20

THEOREM If $\psi:\mathcal{I}\rightarrow\mathbb{R}_{\geq 0}$ is nonnegative and quasi subadditive on $S$ and if

\text{BC}\int\limits_{S}\psi

is finite, then $\psi$ is quasi additive on $S$ .

PROOF To simplify the combinatorics, take $S=A$ . Since

\text{BC}\ \int\limits_{A}\psi

exists and is finite, given $\varepsilon>0$ there is a number $\mu(\varepsilon)>0$ such that for any $D_{0}=[I_{0}]\in\mathcal{D}$ with $\delta(D_{0})<\mu(\varepsilon)$ , we have

\bigg{|}\text{BC}\ \int\limits_{A}\ \psi-\sum\limits_{I_{0}}\psi(I_{0})\bigg{|}\ <\ \frac{\varepsilon}{3},

where $\displaystyle\sum\limits_{I_{0}}$ is a sum ranging over all $I_{0}\in D_{0}$ . Now choose $D_{0}$ in such a way that

\delta(D_{0})\ <\ \min\{\mu(\varepsilon),\eta(\varepsilon/6)\},

take

\lambda^{\prime}(\varepsilon)\ =\ \min\{\mu(\varepsilon),\lambda(\varepsilon/6,D_{0})\},

and consider any system $D=[I]$ with $\delta(D)<\lambda^{\prime}$ . Since $\psi$ is quasi subadditive,

\sum\limits_{I_{0}}\bigg{[}\sum\limits_{I\subset I_{0}}\ \psi(I)-\psi(I_{0})\bigg{]}^{-}\ <\ \frac{\varepsilon}{6}.

On the other hand,

\bigg{|}\text{BC}\int\limits_{A}\ \psi-\sum\limits_{I}\psi(I)\bigg{|}\ <\ \frac{\varepsilon}{3}.

Denote by $\displaystyle\sum^{\prime}$ a sum over all $I\in D$ with $I\not\subset I_{0}$ for any $I_{0}\in D_{0}$ $-$ then

	$\displaystyle 0\$	$\displaystyle\leq\ \sum\limits_{I_{0}}\ \bigg{\|}\sum\limits_{I\subset I_{0}}\ \psi(I)-\psi(I_{0})\bigg{\|}\ +\ \sum{}^{\prime}\ \psi(I)$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \bigg{[}\sum\limits_{I\subset I_{0}}\ \psi(I)-\psi(I_{0})\bigg{]}+2\hskip 1.42271pt\sum\limits_{I_{0}}\bigg{[}\sum\limits_{I\subset I_{0}}\psi(I)-\psi(I_{0})\bigg{]}^{-}$
		$\displaystyle\hskip 56.9055pt+\sum{}^{\prime}\psi(I)$
		$\displaystyle=\ \bigg{[}\sum\limits_{I}\psi(I)-\text{BC}\int\limits_{A}\psi\bigg{]}-\bigg{[}\sum\limits_{I_{0}}\ \psi(I_{0})-\text{BC}\int\limits_{A}\psi\bigg{]}$
		$\displaystyle\hskip 56.9055pt+2\hskip 1.42271pt\sum\limits_{I_{0}}\hskip 1.42271pt\bigg{[}\sum\limits_{I\subset I_{0}}\psi(I)-\psi(I_{0})\bigg{]}^{-}$
		$\displaystyle\leq\ \frac{\varepsilon}{3}\ +\ \frac{\varepsilon}{3}\ +\ 2\hskip 1.42271pt\frac{\varepsilon}{6}$
		$\displaystyle=\ \varepsilon.$

The requirements for quasi additivity are thus met.

myn21

THEOREM Suppose that $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ is quasi additive on $S$ $-$ then $\left\lVert\phi\right\rVert:\mathcal{I}\rightarrow\mathbb{R}_{\geq 0}$ is quasi additive on $S$ .

PROOF Fix $\varepsilon>0$ , take $S=A$ , and in the notation above, introduce $\eta(\varepsilon)$ , $D_{0}=[I_{0}]$ , $\lambda(\varepsilon,D_{0})$ , $D=[I]$ $-$ then the objective is to show that

\sum\limits_{I_{0}}\ \bigg{[}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\bigg{]}^{-}\ <\ \varepsilon.

To this end, let

\Phi(I_{0})\ =\ \sum\limits_{I\subset I_{0}}\ \phi(I)-\phi(I_{0}).

Then

	$\displaystyle\left\lVert\phi(I_{0})+\Phi(I_{0})\right\rVert$	$\displaystyle=\ \sum\limits_{I\subset I_{0}}\ \phi(I)\$
		$\displaystyle=\ \bigg{[}\sum\limits_{m=1}^{M}\ \bigg{(}\sum\limits_{I\subset I_{0}}\ \phi_{m}(I)\bigg{)}^{2}\ \bigg{]}^{\frac{1}{2}}$
		$\displaystyle\leq\ \sum\limits_{I\subset I_{0}}\ \bigg{[}\sum\limits_{m=1}^{M}\ \phi_{m}(I)^{2}\ \bigg{]}^{\frac{1}{2}}$
		$\displaystyle=\ \sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert.$

Meanwhile

\phi(I_{0})=[\phi(I_{0})+\Phi(I_{0})]+[-\Phi(I_{0})]

$\implies$

	$\displaystyle\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\$	$\displaystyle\geq\ \left\lVert\phi(I_{0})+\Phi(I_{0})\right\rVert-\left\lVert\phi(I_{0})\right\rVert$
		$\displaystyle\geq-\left\lVert\Phi(I_{0})\right\rVert$

$\implies$

\bigg{[}\sum\limits_{I\subset I_{0}}\left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\bigg{]}^{-}\ \leq\ \left\lVert\Phi(I_{0})\right\rVert

$\implies$

	$\displaystyle\sum\limits_{I_{0}}\ \bigg{[}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\bigg{]}^{-}\$	$\displaystyle\leq\ \sum\limits_{I_{0}}\ \left\lVert\Phi(I_{0})\right\rVert$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \big{\|}\big{\|}\sum\limits_{I\subset I_{0}}\ {\phi(I)}-{\phi(I_{0})}\big{\|}\big{\|}$
		$\displaystyle<\ \varepsilon,$

$\phi$ being quasi additive.

myn22

APPLICATION If $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ is quasi additive, then the interval functions

I\rightarrow\left|\phi_{m}(I)\right|\quad(m=1,\ldots,M)

are quasi subadditive.

[In fact, the quasi additivity of the $\phi$ implies the quasi additivity of the $\phi_{m}$ and

\left\lVert\phi_{m}\right\rVert=\left|\phi_{m}\right|.]

[Note: It is also true that $\phi_{m}^{+},\ \phi_{m}^{-}$ are quasi subadditive.]

myn23

LEMMA If $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ is quasi additive on $S$ and if

\text{BC}\int\limits_{S}\ \left\lVert\phi\right\rVert\ <\ +\infty,

then $\phi$ is quasi additive on every subset $S^{\prime}\subset S$ .

PROOF First of all, $\left\lVert\phi\right\rVert$ is quasi subadditive on $S$ , hence also on $S^{\prime}$ . Therefore

\text{BC}\int\limits_{S^{\prime}}\ \left\lVert\phi\right\rVert

exists and

\text{BC}\int\limits_{S^{\prime}}\ \left\lVert\phi\right\rVert\ \leq\ \text{BC}\int\limits_{S}\ \left\lVert\phi\right\rVert\ <\ +\infty,

from which it follows that $\left\lVert\phi\right\rVert$ is quasi additive on $S^{\prime}$ . Given $\varepsilon>0$ , determine the parameters in the definition of quasi additive in such a way that the relevant relations are simultaneously satisfied per $\phi$ on $S$ and per $\left\lVert\phi\right\rVert$ on $S^{\prime}$ , hence

	$\displaystyle\sum\limits_{I_{0}}s(I_{0},S^{\prime})\ \big{\|}\big{\|}\sum\limits_{I}s(I,I_{0})\ \phi(I)-\phi(I_{0})\big{\|}\big{\|}\$	$\displaystyle\leq\ \sum\limits_{I_{0}}s(I_{0},S)\ \big{\|}\big{\|}\sum\limits_{I}s(I,I_{0})\ \phi(I)-\phi(I_{0})\big{\|}\big{\|}$
		$\displaystyle<\ \varepsilon$

and

\sum\limits_{I}s(I,S^{\prime})\ \bigg{[}1-\sum\limits_{I_{0}}\ s(I,I_{0})\hskip 0.85355pts(I_{0},S^{\prime})\bigg{]}\left\lVert\phi(I)\right\rVert\ <\ \varepsilon.

Therefore $\phi$ is quasi additive on $S^{\prime}$ .

myn24

APPLICATION If $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ is quasi additive and if

\text{BC}\int\limits_{A}\ \left\lVert\phi\right\rVert\ <\ +\infty,

then $\phi$ is quasi additive on every subset of $A$ .

Here is a summary of the fundamental points of this $\S$ . Work with $\phi$ and $\left\lVert\phi\right\rVert$ .

• Suppose that $\left\lVert\phi\right\rVert$ is quasi subadditive on $S$ and

\text{BC}\int\limits_{S}\ \left\lVert\phi\right\rVert\ <\ +\infty.

Then $\left\lVert\phi\right\rVert$ is quasi additive on $S$ .

• Suppose that ${\phi}$ is quasi additive on $S$ $-$ then $\left\lVert\phi\right\rVert$ is quasi subadditive on $S$ .

So $:$ If $\phi$ is quasi additive on $S$ AND if

\text{BC}\int\limits_{S}\ \left\lVert\phi\right\rVert\ <\ +\infty,

then $\left\lVert\phi\right\rVert$ is quasi additive on $S$ .

[Note: It is not true in general that $\left\lVert\phi\right\rVert$ quasi additive implies $\phi$ quasi additive.]

myn25

EXAMPLE Take $A=[a,b]$ and let $\mathcal{I},\ \mathcal{D}$ , and $\delta$ be as at the beginning. Given a continuous curve

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M},

define a quasi additive interval function $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ by the rule

	$\displaystyle\phi(I)\$	$\displaystyle=\ (\phi_{1}(I),\ldots,\phi_{M}(I))$
		$\displaystyle=\ (f_{1}(d)-f_{1}(c),\ldots,f_{M}(d)-f_{M}(c)),$

where $I=[c,d]\subset[a,b]$ , thus

\left\lVert\phi(I)\right\rVert\ =\ \left\lVert\underline{f}(d)-\underline{f}(c)\right\rVert,

so if $P\in\mathcal{P}[a,b]$ corresponds to

D\longleftrightarrow\{[x_{i-1},x_{i}]:i=1,\ldots,n\},

then

\sum\limits_{I\in D}\ \left\lVert\phi(I)\right\rVert\ =\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert

$\implies$

	$\displaystyle\text{BC}\int\limits_{A}\left\lVert\phi\right\rVert\$	$\displaystyle=\ \lim\limits_{\delta(D)\rightarrow 0}\ \sum\limits_{I\in D}\ \left\lVert\phi(I)\right\rVert$
		$\displaystyle=\ \lim\limits_{\left\lVert P\right\rVert\rightarrow 0}\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert$
		$\displaystyle=\ \ell(C).$

Therefore $C$ is rectifiable iff

\text{BC}\int\limits_{A}\left\lVert\phi\right\rVert\ <\ +\infty.

And when this is the case, $\left\lVert\phi\right\rVert$ is quasi additive on $A$ .

[Note: A priori,

\ell(C)\ =\ \sup\limits_{P\in\mathcal{P}[a,b]}\ \sum\limits_{i=1}^{n}\ \left\lVert\underline{f}(x_{i})-\underline{f}(x_{i-1})\right\rVert.

But here, thanks to the continuity of $\underline{f}$ , the $\sup$ can be replace by $\lim$ .]

myn26

EXAMPLE Take $A=[a,b]$ and let $\mathcal{I}$ and $\mathcal{D}$ be as above. Suppose that

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}

is a rectifiable curve, potentially discontinuous.

• Given $a\leq x_{0}<b$ , put

s^{+}(x_{0})\ =\ \limsup\limits_{x\downarrow x_{0}}\ \left\lVert\underline{f}(x)-\underline{f}(x_{0})\right\rVert

and let $s^{+}(b)=0.$

• Given $a<x_{0}\leq b$ , put

s^{-}(x_{0})\ =\ \limsup\limits_{x\uparrow x_{0}}\left\lVert\underline{f}(x)-\underline{f}(x_{0})\right\rVert

and let $s^{-}(a)=0$ . Combine the data and set

s(x)\ =\ s^{+}(x)+s^{-}(x)\qquad(a\leq x\leq b).

Then $s(x)$ is zero everywhere save for at most countably many $x$ and

\sigma\ =\ \sum\limits_{x}s(x)\ \leq\ \ell(C).

Take $\phi$ as above and define a mesh $\delta$ by the rule

\delta(D)\ =\ \left\lVert P\right\rVert+\sigma-\sum\limits_{i=0}^{n}s(x_{i}).

One can then show that $\phi$ is quasi additive and

\text{BC}\int\limits_{A}\left\lVert\phi\right\rVert\ =\ \ell(C).

myn27

NOTATION Given a quasi additive interval function $\phi$ , let

V[\phi,S]\ =\ \sup\limits_{D\in\mathcal{D}}\ \sum\ [\hskip 0.85355pt\left\lVert\phi\right\rVert,\hskip 0.85355ptS,\hskip 0.85355ptD\hskip 0.85355pt].

myn28

N.B. By definition,

\text{BC}\int\limits_{S}\ \left\lVert\phi\right\rVert\ =\ \lim\limits_{\delta(D)\rightarrow 0}\ \sum\ [\hskip 0.85355pt\left\lVert\phi\right\rVert,\hskip 0.85355ptS,\hskip 0.85355ptD\hskip 0.85355pt],

\text{BC}\int\limits_{S}\left\lVert\phi\right\rVert\ \leq\ V[\phi,S]

and strict inequality may hold.

myn29

LEMMA Given a quasi additive $\phi$ and a subset $S\subset A$ , suppose that for every $\varepsilon>0$ and any $D_{0}=[I_{0}]$ there exists $\lambda(\varepsilon,S,D_{0})>0$ such that for every system $D=[I]$ with $\delta(D)<\lambda(\varepsilon,S,D_{0})$ the relation

\sum\limits_{I_{0}}s(I_{0},S)\ \bigg{[}\sum\limits_{I}s(I,I_{0})\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\bigg{]}^{-}\ <\ \varepsilon

obtains $-$ then

\text{BC}\int\limits_{S}\ \left\lVert\phi\right\rVert\ =\ V[\phi,S].

$\boldsymbol{\S}$ 10. LINE INTEGRALS (bis)

Throughout this $\S$ , the situation will be absolute, where $A=[a,b]$ and $\mathcal{I},\ \mathcal{D}$ , and $\delta$ have their usual connotations.

C\longleftrightarrow\underline{f}:[a,b]\rightarrow\mathbb{R}^{M}

is a curve, continuous and rectifiable, then

\text{BC}\ \int\limits_{A}\ \left\lVert\phi\right\rVert\ =\ \ell(C).

And if $F$ is a parametric integrand, then

\int\limits_{C}F\ =\ \lim\limits_{\left\lVert P\right\rVert\rightarrow 0}\ \sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))

exists, the result being independent of the $\xi_{i}$ .

myn1

N.B. Recall the procedure: Introduce the integral

I(C)\ =\ \int\limits_{a}^{b}\ F(\underline{f}(x),\underline{\theta}(x))\ \text{d}\mu_{C}

and prove that

\lim\limits_{\left\lVert P\right\rVert\rightarrow 0}\ \sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))

exists and equals $I(C)$ , the result being denoted by the symbol

\int\limits_{C}\ F

and called the line integral of $F$ along $C$ .

There is another approach to all of this which does not use measure theory. Thus define an interval function $\Phi:\mathcal{I}\rightarrow\mathbb{R}$ by the prescription

\Phi(I;\xi)\ =\ F(\underline{f}(\xi),\phi(I)),

where $\xi\in I$ is arbitrary.

[Note: By definition

\phi(I)=(\phi_{1}(I),\ldots,\phi_{M}(I))\ =\ (f_{1}(d)-f_{1}(c),\ldots,f_{M}(d)-f_{M}(c)),

$I$ being $[c,d]\subset[a,b]$ . Moreover, $\phi$ is quasi additive.]

myn2

THEOREM $\Phi$ is quasi additive.

Admit the contention $-$ then

\lim\limits_{\delta(D)\rightarrow 0}\ \sum\limits_{I\in D}\ \Phi(I;\xi)\ =\ \lim\limits_{\left\lVert P\right\rVert\rightarrow 0}\ \sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))

exists, call it

(\xi)\ \int\limits_{C}F.

myn3

N.B. Needless to say, it turns out that

(\xi)\ \int\limits_{C}F

is independent of the $\xi$ $($ this follows by a standard `` $\varepsilon/3$ '' argument $)$ $($ details at the end $)$ .

[Note: This is one advantage of the approach via $I(C)$ in that independence is manifest.]

To simplify matters, it will be best to generalize matters.

Assume from the outset that $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ is now an arbitrary interval function which is quasi additive with

\text{BC}\ \int\limits_{A}\left\lVert\phi\right\rVert\ <\ +\infty,

hence that $\left\lVert\phi\right\rVert$ is also quasi additive as well.

Introduce another interval function $\zeta:\mathcal{I}\rightarrow\mathbb{R}^{N}$ and expand the definition of parametric integrand so that

F:X\times\mathbb{R}^{M}\rightarrow\mathbb{R},

where $X\subset\mathbb{R}^{N}$ is compact and $\zeta(I)\subset X$ .

myn4

EXAMPLE To recover the earlier set up, take $N=M$ , keep $\phi:\mathcal{I}\rightarrow\mathbb{R}^{M}$ , let $\omega:\mathcal{I}\rightarrow[a,b]$ be a choice function, i.e., suppose that $\omega(I)\in I\subset[a,b]$ , let $\zeta(I)=\underline{f}(\omega(I))$ , and take $X=[C]\subset\mathbb{R}^{M}$ .

myn5

CONDITION $(\zeta)$ $\forall\ \varepsilon>0$ , $\exists\ t(\varepsilon)>0$ such that if $D_{0}=[I_{0}]$ is any system subject to $\delta(D_{0})<t(\varepsilon)$ there also exists $T(\varepsilon,D_{0})$ , such that for any system $D=[I]$ with $\delta(D)<T(\varepsilon,D_{0})$ the relation

\max\limits_{I_{0}}\ \max\limits_{I\subset I_{0}}\ \left\lVert\zeta(I)-\zeta(I_{0})\right\rVert\ <\ \varepsilon

obtains.

myn6

N.B. Owing to the uniform continuity of $\underline{f}$ , this condition is automatic in the special case supra.

myn7

THEOREM Let $F$ be a parametric integrand, form the interval function $\Phi:\mathcal{I}\rightarrow\mathbb{R}$ defined by the prescription

\Phi(I)\ =\ F(\zeta(I),\phi(I)),

and impose condition $(\zeta)$ $-$ then $\Phi$ is quasi additive.

The proof will emerge from the discussion below but there are some preliminaries that have to be dealt with first.

Start by writing down simultaneously ( $\text{qa}_{1}$ - $A$ ) and ( $\text{qa}_{2}$ - $A$ ) for $\phi$ and $\left\lVert\phi\right\rVert$ (both are quasi additive), $\bar{\varepsilon}$ to be determined.

\begin{cases}\displaystyle\ \sum\limits_{I_{0}}\ \bigg{|}\bigg{|}\sum\limits_{I\subset I_{0}}\ \phi(I)-\phi(I_{0})\bigg{|}\bigg{|}\ <\ \bar{\varepsilon}\\[26.0pt] \displaystyle\ \sum\limits_{I\not\subset I_{0}}\ \left\lVert\phi(I)\right\rVert\ <\ \bar{\varepsilon}\end{cases}\quad\begin{cases}\displaystyle\ \sum\limits_{I_{0}}\ \bigg{|}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\bigg{|}\ <\ \bar{\varepsilon}\\[26.0pt] \displaystyle\ \sum\limits_{I\not\subset I_{0}}\ \left|\left\lVert\phi(I)\right\rVert\right|\ <\ \bar{\varepsilon}\end{cases}

for $\delta(D_{0})<\eta(\bar{\varepsilon})$ and $\delta(D)<\lambda(\bar{\varepsilon},D_{0})$ and in addition

\bigg{|}\sum\limits_{I\in D}\ \left\lVert\phi(I)\right\rVert-\text{BC}\ \int\limits_{A}\ \left\lVert\phi(I)\right\rVert\bigg{|}\ <\ \bar{\varepsilon}

for $\delta(D)<\sigma(\bar{\varepsilon})$ .

Fix $\varepsilon>0$ . Put

V\ =\ \text{BC}\ \int\limits_{A}\left\lVert\phi\right\rVert\qquad(<+\infty).

• $(F)$ $X\times\text{U}(M)$ is a compact set on which $F$ is bounded:

\left|F(\underline{x},\underline{t})\right|\ \leq\ C\qquad(\underline{x}\in X,\hskip 0.85355pt\underline{t}\in\text{U}(M))

and uniformly continuous: $\exists\ \gamma$ such that

\begin{cases}\left\lVert\underline{x}-\underline{x}^{\prime}\right\rVert\\[8.0pt] \left\lVert\underline{t}-\underline{t}^{\prime}\right\rVert\end{cases}<\ \gamma\ \implies\ \left|F(\underline{x},\underline{t})-F(\underline{x}^{\prime},\underline{t}^{\prime})\right|\ <\ \frac{\varepsilon}{3(V+\varepsilon)}.

• $(\alpha)$

\alpha(I_{0})\ =\ \frac{\phi(I_{0})}{\left\lVert\phi(I_{0})\right\rVert}\quad\text{if }\phi(I_{0})\neq 0

but 0 otherwise and

\alpha(I)\ =\ \frac{\phi(I)}{\left\lVert\phi(I)\right\rVert}\quad\text{if }\phi(I)\neq 0

but 0 otherwise.

myn8

NOTATION Denote by

\sum{}_{\gamma^{+}}^{(I_{0})}

the sum over the $I\subset I_{0}$ for which

\left\lVert\alpha(I_{0})-\alpha(I)\right\rVert\ \geq\ \gamma

and denote by

\sum{}_{\gamma^{-}}^{(I_{0})}

the sum over the $I\subset I_{0}$ for which

\left\lVert\alpha(I_{0})-\alpha(I)\right\rVert\ <\ \gamma.

Therefore

\sum\limits_{I\subset I_{0}}\ =\ \sum{}_{\gamma^{+}}^{(I_{0})}\ +\ \sum{}_{\gamma^{-}}^{(I_{0})}.

myn9

LEMMA

\displaystyle\frac{\gamma^{2}}{2}\ \sum\limits_{I_{0}}\ \sum{}_{\gamma^{+}}^{(I_{0})}\ \left\lVert\phi(I)\right\rVert\ \leq\ \

\displaystyle\sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\ \phi(I)-\phi(I_{0})\Big{\|}\ +\ \sum\limits_{I_{0}}\ \Big{|}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{|}.

PROOF The inequality

\left\lVert\alpha(I_{0})-\alpha(I)\right\rVert\ \geq\ \gamma

implies that

	$\displaystyle\gamma^{2}\$	$\displaystyle\leq\ \left\lVert\alpha(I_{0})-\alpha(I)\right\rVert^{2}$
		$\displaystyle=\ (\alpha(I_{0})-\alpha(I))\cdot(\alpha(I_{0})-\alpha(I))$
		$\displaystyle=\ \left\lVert\alpha(I_{0})\right\rVert^{2}-2\alpha(I_{0})\cdot\alpha(I)+\left\lVert\alpha(I)\right\rVert^{2}$
		$\displaystyle=2-2\alpha(I_{0})\cdot\alpha(I),$

\frac{\gamma^{2}}{2}\ \leq\ 1-\alpha(I_{0})\cdot\alpha(I)

$\implies$

\frac{\gamma^{2}}{2}\left\lVert\phi(I)\right\rVert\ \leq\ \left\lVert\phi(I)\right\rVert-\alpha(I_{0})\cdot\phi(I).

But for any $I$ ,

0\ \leq\ \left\lVert\phi(I)\right\rVert-\alpha(I_{0})\cdot\phi(I).

Proof: In fact,

\left\lVert\phi(I)\right\rVert-\frac{\phi(I_{0})\cdot\phi(I)}{\left\lVert\phi(I_{0})\right\rVert}\ =\ \frac{1}{\left\lVert\phi(I_{0})\right\rVert}\left[\left\lVert\phi(I)\right\rVert\left\lVert\phi(I_{0})\right\rVert-\phi(I_{0})\cdot\phi(I)\right].

Now quote Schwarz's inequality. Thus we may write

	$\displaystyle\frac{\gamma^{2}}{2}\ \sum{}_{\gamma^{+}}^{(I_{0})}\left\lVert\phi(I)\right\rVert\$	$\displaystyle\leq\ \sum{}_{\gamma^{+}}^{(I_{0})}\ (\left\lVert\phi(I)-\alpha(I_{0})\cdot\phi(I))\right\rVert$
		$\displaystyle\leq\ \sum\limits_{I\subset I_{0}}\ (\left\lVert\phi(I)-\alpha(I_{0})\cdot\phi(I))\right\rVert$
		$\displaystyle=\ \sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert+\alpha(I_{0})\cdot\bigl{(}\phi(I_{0})-\sum\limits_{I\subset I_{0}}\ \phi(I)\bigr{)}$
		$\displaystyle\leq\ \bigg{\|}\sum\limits_{I\subset I_{0}}\ (\left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\bigg{\|}+\big{\|}\big{\|}\phi(I_{0})-\sum\limits_{I\subset I_{0}}\ \phi(I)\big{\|}\big{\|}\qquad(\text{Schwarz}).$

To finish, sum over $I_{0}$ .

• $(D_{0})$ Assume

\qquad\qquad\delta(D_{0})\ <\ \min\{t(\gamma),\hskip 0.85355pt\eta(\varepsilon),\hskip 0.85355pt\eta(\varepsilon\gamma^{2})\}.

• $(D)$ Assume

\qquad\qquad\delta(D)\ <\ \min\{\sigma(\varepsilon),\hskip 0.85355pt\lambda(\varepsilon,D_{0}),\hskip 0.85355pt\lambda(\varepsilon\gamma^{2},D_{0}),\hskip 0.85355ptT(\gamma,D_{0})\}.

• $(\bar{\varepsilon})$ Assume

\qquad\qquad\bar{\varepsilon}\ <\ \min\left\{\gamma,\hskip 0.85355pt\frac{\varepsilon}{3C},\hskip 0.85355pt\frac{\varepsilon\gamma^{2}}{24\hskip 0.85355ptC}\right\}.

Then

	$\displaystyle\sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\$	$\displaystyle\Phi(I)-\Phi(I_{0})\Big{\|}\$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\ F(\zeta(I),\phi(I))-\sum\limits_{I\subset I_{0}}\ F(\zeta(I_{0}),\alpha(I_{0}))\left\lVert\phi(I)\right\rVert$
		$\displaystyle\qquad\qquad+\sum\limits_{I\subset I_{0}}\ F(\zeta(I_{0}),\alpha(I_{0}))\left\lVert\phi(I)\right\rVert-F(\zeta(I_{0}),\alpha(I_{0}))\left\lVert\phi(I_{0})\right\rVert\Big{\|}$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\ F(\zeta(I),\alpha(I))-F(\zeta(I_{0}),\alpha(I_{0})))\left\lVert\phi(I)\right\rVert$
		$\displaystyle\qquad\qquad+\sum\limits_{I\subset I_{0}}\ F(\zeta(I_{0}),\alpha(I_{0}))\hskip 0.85355pt(\left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert)\Big{\|}$
		$\displaystyle\leq\ \sum\limits_{I_{0}}\Big{\|}F(\zeta(I_{0}),\alpha(I_{0}))\Big{\|}\quad\Big{\|}\sum\limits_{I\subset I_{0}}\left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{\|}$
		$\displaystyle\qquad\qquad+\sum\limits_{I_{0}}\ \sum\limits_{I\subset I_{0}}\ \|F(\zeta(I),\alpha(I))-F(\zeta(I_{0}),\alpha(I_{0}))\|\ \left\lVert\phi(I)\right\rVert$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \Big{\|}F(\zeta(I_{0}),\alpha(I_{0}))\Big{\|}\quad\Big{\|}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{\|}$
		$\displaystyle\qquad+\sum\limits_{I_{0}}\ \Big{(}\sum{}_{\gamma^{-}}^{(I_{0})}+\sum{}_{\gamma^{+}}^{(I_{0})}\Big{)}\big{\|}F(\zeta(I_{0}),\alpha(I_{0}))-F(\zeta(I),\alpha(I))\big{\|}\left\lVert\phi(I)\right\rVert.$

First:

	$\displaystyle\sum\limits_{I_{0}}\Big{\|}F(\zeta(I_{0}),\alpha(I_{0}))\Big{\|}\ \hskip 1.42271pt\Big{\|}\sum\limits_{I\subset I_{0}}\left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{\|}\$	$\displaystyle\leq\ C\hskip 1.42271pt\sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{\|}$
		$\displaystyle<\ C\bar{\varepsilon}$
		$\displaystyle<\ C\frac{\varepsilon}{3C}$
		$\displaystyle=\ \frac{\varepsilon}{3}.$

Second: Consider

\sum\limits_{I_{0}}\ \sum{}_{\gamma^{-}}^{(I_{0})}\ \big{|}F(\zeta(I_{0}),\alpha(I_{0}))-F(\zeta(I),\alpha(I))\big{|}\left\lVert\phi(I)\right\rVert.

Here

\begin{cases}\left\lVert\alpha(I_{0})\right\rVert=1,\\[8.0pt] \left\lVert\alpha(I)\right\rVert=1,\\[8.0pt] \left\lVert\alpha(I_{0})-\alpha(I)\right\rVert<\gamma,\\[8.0pt] \left\lVert\zeta(I_{0})-\zeta(I)\right\rVert<\gamma\end{cases}\implies\big{|}F(\zeta(I_{0}),\alpha(I_{0}))-F(\zeta(I),\alpha(I))\big{|}\ <\ \frac{\varepsilon}{3(V+\varepsilon)}.

The entity in question is thus majorized by

	$\displaystyle\frac{\varepsilon}{3(V+\varepsilon)}\ \sum\limits_{I_{0}}\ \sum{}_{\gamma^{-}}^{(I_{0})}\left\lVert\phi(I)\right\rVert\$	$\displaystyle\leq\ \frac{\varepsilon}{3(V+\varepsilon)}\ \sum\limits_{I\in D}\ \left\lVert\phi(I)\right\rVert$
		$\displaystyle\leq\ \frac{\varepsilon}{3(V+\varepsilon)}(V+\varepsilon)$
		$\displaystyle=\ \frac{\varepsilon}{3}.$

Third:

	$\displaystyle\sum\limits_{I_{0}}\ \sum{}_{\gamma^{+}}^{(I_{0})}\ \big{\|}$	$\displaystyle F(\zeta(I_{0}),\alpha(I_{0}))-F(\zeta(I),\alpha(I))\big{\|}\left\lVert\phi(I)\right\rVert$
		$\displaystyle\leq\ 2\hskip 0.85355ptC\ \sum\limits_{I_{0}}\ \sum{}_{\gamma^{+}}^{(I_{0})}\ \left\lVert\phi(I)\right\rVert$
		$\displaystyle\leq\ \frac{4\hskip 0.85355ptC}{\gamma^{2}}\ \Big{[}\sum\limits_{I_{0}}\ \sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)-\phi(I_{0})\right\rVert\ +\ \sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{\|}\hskip 1.42271pt\Big{]}$
		$\displaystyle\leq\ \frac{4\hskip 0.85355ptC}{\gamma^{2}}(\bar{\varepsilon}+\bar{\varepsilon})$
		$\displaystyle=\ \frac{8\hskip 0.85355ptC}{\gamma^{2}}\bar{\varepsilon}$
		$\displaystyle<\ \frac{8\hskip 0.85355ptC}{\gamma^{2}}\cdot\frac{\varepsilon\gamma^{2}}{24\hskip 0.85355ptC}$
		$\displaystyle=\ \frac{\varepsilon}{3}.$

In total then:

	$\displaystyle\sum\limits_{I_{0}}\Big{\|}\sum\limits_{I\not\subset I_{0}}\ \Phi(I)-\Phi(I_{0})\Big{\|}\$	$\displaystyle<\ \frac{\varepsilon}{3}\hskip 1.42271pt+\hskip 1.42271pt\frac{\varepsilon}{3}\hskip 1.42271pt+\hskip 1.42271pt\frac{\varepsilon}{3}$
		$\displaystyle=\ \varepsilon.$

And finally

	$\displaystyle\sum\limits_{I\not\subset I_{0}}\ \left\|\Phi(I)\right\|\$	$\displaystyle=\ \sum\limits_{I\not\subset I_{0}}\ \left\|F(\zeta(I),\phi(I))\right\|$
		$\displaystyle=\ \sum\limits_{I\not\subset I_{0}}\ \left\|F(\zeta(I),\alpha(I))\right\|\left\lVert\phi(I)\right\rVert$
		$\displaystyle\leq\ C\hskip 1.42271pt\sum\limits_{I\not\subset I_{0}}\ \left\lVert\phi(I)\right\rVert$
		$\displaystyle<\ C\hskip 1.42271pt\bar{\varepsilon}$
		$\displaystyle<\ C\hskip 1.42271pt\frac{\varepsilon}{3\hskip 0.85355ptC}$
		$\displaystyle=\ \frac{\varepsilon}{3}$
		$\displaystyle<\ \varepsilon.$

Therefore $\Phi$ is quasi additive. And since the conditions on $F$ carry over to $\left|F\right|$ , it follows that $\left\lVert\Phi\right\rVert$ is also quasi additive, hence

\text{BC}\ \int\limits_{A}\ \left\lVert\Phi\right\rVert

exists and is finite.

To tie up one loose end, return to the beginning and consider the line integrals

(\xi)\ \int\limits_{C}\ F,\qquad(\xi^{\prime})\ \int\limits_{C}\ F,

the claim being that they are equal. That this is so can be seen by writing

	$\displaystyle\Big{\|}(\xi)\ \int\limits_{C}\ F\ \ -\ \$	$\displaystyle(\xi^{\prime})\ \int\limits_{C}\ F\ \Big{\|}\$
		$\displaystyle=\ \Big{\|}(\xi)\int\limits_{C}\ F-\sum\limits_{i=1}^{n}F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))$
		$\displaystyle\quad\qquad+\sum\limits_{i=1}^{n}F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))$
		$\displaystyle\quad\qquad-\sum\limits_{i=1}^{n}F(\underline{f}(\xi_{i}^{\prime}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))$
		$\displaystyle\quad\qquad+\sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}^{\prime}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))\ -\ (\xi^{\prime})\int\limits_{C}\ F\ \Big{\|}$

and proceed from here in the obvious way.

myn10

EXAMPLE Take $N=1$ , $M=1$ and define an interval function $\left|\ \cdot\ \right|:\mathcal{I}\rightarrow\mathbb{R}$ by sending $I$ to its length $\left|I\right|$ . Fix a choice function $\omega:\mathcal{I}\rightarrow[a,b]$ . Consider a curve

C\longleftrightarrow f:[a,b]\rightarrow\mathbb{R}.

Assume: $f$ is continuous and of bounded variation, thus

\ell(C)\ =\ T_{f}[a,b]\ <\ +\infty.

Work with the parametric integrand $F(x,t)=xt$ $-$ then the data

	$\displaystyle I$	$\displaystyle\rightarrow F(\zeta(I),\left\|I\right\|)$
		$\displaystyle=\ F(f(\omega(I)),\left\|I\right\|)$
		$\displaystyle=\ f(\omega(I))\left\|I\right\|$

leads to sums of the form

\sum\limits_{i=1}^{n}\ f(\xi_{i})(x_{i}-x_{i-1}),

hence to

\int\limits_{C}F\ =\ \int\limits_{a}^{b}f,

the Riemann integral of $f$ .

$\boldsymbol{\S}$ 11. EXAMPLES

myn1

EXAMPLE Suppose that $A=[0,1]$ with the usual topology, $\{I\}$ the class of all intervals $I=[a,b]\subset A$ , $a<b$ , and define $\left|I\right|=b-a$ . Let $\mathcal{D}$ be the family of all finite systems $D=[I]$ of nonoverlapping $I\in\{I\}$ . If we take

\delta(D)\ =\ \left(1-\sum\ \left|I\right|\right)+\max\left|I\right|,

then

0\ <\ \delta(D)<2,

and the Assumption §9, #4, is trivially satisfied.

Take $k=1$ and $\phi(I)=b-a>0$ . Given $\varepsilon>0$ arbitrary take $\eta=\eta(\varepsilon)=\displaystyle\frac{\varepsilon}{2}$ and let

D_{0}\ =\ (I_{j},\ j=1,\ldots,N)

be any system $D_{0}\in\mathcal{D}$ with $\delta(D_{0})<\eta$ . Take

\lambda\ =\ \lambda(\varepsilon,D_{0})\ =\ \frac{\varepsilon}{4N}.

If $D=[J]$ is any system with $\delta(D)<\lambda$ , and we denote by $\sum{}^{\prime}$ any sum ranging over all $J\in D$ with $J\not\subset I$ and for any $I\in D_{0}$ , we have

	$\displaystyle\sum{}^{\prime}\ \phi(J)\$	$\displaystyle<\ \left(1-\sum\left\|I_{j}\right\|\right)+2\hskip 0.85355ptN(\varepsilon/4N)$
		$\displaystyle<\ \frac{\varepsilon}{2}+\frac{\varepsilon}{2}$
		$\displaystyle=\ \varepsilon,$

and $(\phi_{2})$ holds. Also, if $\sum{}^{(j)}$ denotes a sum ranging over all $J\in D$ with $J\subset I_{j}$ , we have

\phi(I_{j})-\sum{}^{(j)}\ \phi(J)\ >\ 0,

and

	$\displaystyle\sum\limits_{j}\ \left(\phi(I_{j})-\sum{}^{(j)}\ \phi(J)\right)\$	$\displaystyle<\ \left(1-\sum\left\|J\right\|\right)+2\hskip 0.85355ptN(\varepsilon/4N)$
		$\displaystyle<\ \frac{\varepsilon}{2}+\frac{\varepsilon}{2}$
		$\displaystyle=\ \varepsilon,$

and $(\phi_{1})$ holds. Thus, $\phi$ is quasi additive with $\mathcal{B}=1$ .

myn2

\delta(D)\ =\ \begin{cases}\left(1-\sum\ \left|I\right|\right)+\max\left|I\right|\hskip 35.56593pt\text{if for no $I\in D$ we have $a=0$}\\[8.0pt] \left(1-\sum\ \left|I\right|\right)+\max\left|I\right|+1\hskip 14.22636pt\text{if for one $I\in D$ we have $I=[0,b]$}\end{cases}.

If $\eta<1$ , and $\delta(D)<1$ , $D=[I]$ , then there is in $D$ no interval $I=[0,b]$ . The assumptions §9, #4 , are trivially satisfied. By the same reasoning as in the previous example with $\displaystyle\eta=\min\hskip 1.42271pt\left[\frac{\varepsilon}{2},1\right]$ it follows that $\phi$ is quasi additive with $V=1$ .

myn3

\delta(D)\ =\ \begin{cases}\left(1-\sum\ \left|I\right|\right)+\max\left|I\right|+1\hskip 17.07182pt\text{if for no $I\in D$ we have $a=0$}\\[8.0pt] \left(1-\sum\ \left|I\right|\right)+\max\left|I\right|\hskip 35.56593pt\text{if for one $I\in D$ we have $I=[0,b]$}\end{cases}.

If $\eta<1$ , and $\delta(D)<1$ , $D=[I]$ , then there is in $D$ one interval $I=[0,b]$ . The assumptions §9, #4 , are trivially satisfied. By the same reasoning as in #2 we prove that $\phi$ is quasi additive with $V=1$ .

myn4

\sigma(m)\ =\ \begin{cases}\ 0\hskip 17.07182pt\text{if $m$ is irrational}\\ \ \displaystyle\frac{1}{p}\hskip 14.22636pt\text{if}\ m=\frac{p}{q},\ p,q\in\mathbb{Z},\ p\geq 1\hskip 1.42271pt\text{minimum},\ 0\leq q\leq p.\end{cases}

Take $k=1$ and

\phi(I)\ =\ I\ =\ b-a\quad\text{for}\ I=[a,b]\subset A.

For $D=[I_{i}:j=1,\ldots,N]$ , $I_{j}=[a_{j},b_{j}]$ , take

\delta(D)\ =\ \left(1-\sum\ \left|I_{j}\right|\right)+\max\left|I_{j}\right|+\sum\ \sigma(a_{j})+\sum\ \sigma(b_{j})

For $D=[I_{1}]$ , $I_{1}=[0,1]$ , we have

\delta(D)\ =\ 0+1+1+1\ =\ 3;

for $D=[I_{1},I_{2}]$ , $I_{1}=[0,1/2]$ , $I_{2}=[1/2,1]$ , we have

\delta(D)\ =\ 1/2+(1+1/2)+(1/2+1)\ =\ 3+1/2.

Thus $\delta(D)$ does not necessarily decrease by refinement. The Assumption §9, #4 , is trivially satisfied. Given $\varepsilon>0$ let $D=[I_{j}]$ with $I_{j}=[a_{j},b_{j}]$ , all $a_{j}$ , $b_{j}$ irrational with

1-\sum\ (b_{j}-a_{j})\ <\ \frac{\varepsilon}{2},\quad(b_{j}-a_{j})\ <\ \frac{\varepsilon}{2}.

Then

\delta(D)<\varepsilon.

We may also choose an integer $M:\ \begin{cases}\ M\geq 3,\\[4.0pt] \ M>\displaystyle\frac{4}{\varepsilon}\end{cases},M$ prime. Suppose that $a_{j}$ , $b_{j}$ are all of the form $\displaystyle\frac{q}{M^{2}}$ , $q=1,2,\ldots,M^{2}-1$ , and $N=M$ . Namely, we may take

	$\displaystyle a_{1}$	$\displaystyle=\frac{(M-1)}{M}+\frac{1}{M^{2}},$
	$\displaystyle a_{2}$	$\displaystyle=b_{1}=\frac{1}{M}+\frac{1}{M^{2}},$
	$\displaystyle a_{3}$	$\displaystyle=b_{2}=\frac{2}{M}+\frac{1}{M^{2}},$
	$\displaystyle\hskip 28.45274pt\vdots$
	$\displaystyle a_{N}$	$\displaystyle=b_{N-1}=\frac{M-1}{M}+\frac{1}{M^{2}},$
	$\displaystyle b_{N}$	$\displaystyle=1-\frac{1}{M^{2}}.$

Then

	$\displaystyle\delta(D)$	$\displaystyle=\ \frac{2}{M^{2}}\hskip 1.42271pt+\hskip 1.42271pt\frac{1}{M}\hskip 1.42271pt+\hskip 1.42271ptM\hskip 0.85355pt\frac{2}{M^{2}}$
		$\displaystyle<\ \frac{4}{M}$
		$\displaystyle<\ \varepsilon.$

This remark shows that the second part of the Assumption #4, §9, holds. The proof that ( $\text{qa}_{1}$ -A) and ( $\text{qa}_{2}$ -A), (cf. #14, §9), hold is the same as for #1. Thus $\phi$ is quasi additive and $\mathcal{B}=1$ .

myn5

\phi(I)\ =\ \begin{cases}\ \left|I\right|=b-a\hskip 14.22636pt\text{ if both $a$ and $b$ are irrational}\\[4.0pt] \ \ a-b\hskip 42.67912pt\text{otherwise}\end{cases}.

For $D=[I]\in\mathcal{D}$ , $I=[a,b],$ take

\delta(D)\ =\ \begin{cases}\left(1-\sum\ \left|I\right|\right)+\max\left|I\right|\hskip 14.22636pt\text{ if both $a$ and $b$ are irrational}\\[11.0pt] \ 1\hskip 120.92421pt\text{otherwise}\end{cases}.

Obviously the Assumptions §9, #4 are satisified. If $D=[I]$ , $\delta(D)<1$ , then all $a$ and $b$ are irrational. Obviously ( $\text{qa}_{1}$ -A) and ( $\text{qa}_{2}$ -A), (cf. #14 §9), are satisfied. $\phi(I)$ is quasi additive, and $\mathcal{B}=1$ .

myn6

\phi(I)\ =\ \begin{cases}\ \left|I\right|=b-a\hskip 14.22636pt\text{ if both $a$ and $b$ are irrational}\\[4.0pt] \ \ a-b\hskip 42.67912pt\text{otherwise}\end{cases}.

For $D=[I]\in\mathcal{D}$ , $I=[a,b],$ take

\delta(D)\ =\ \begin{cases}\left(1-\sum\ \left|I\right|\right)+\max\left|I\right|\hskip 14.22636pt\text{ if both $a$ and $b$ are rational}\\[11.0pt] \ 1\hskip 120.92421pt\text{otherwise}\end{cases}.

myn7

EXAMPLE (Jordan length for continuous curves)

Let $C:x=x(t)$ , $0\leq t\leq 1$ , $x=(x_{1},\ldots,x_{k})$ , be any real continuous vector function (a continuous parameteric curve). Take $A=[0,1]$ , $\{I\}$ the class of all intervals $I=[a,b]\subset A$ , $\mathcal{D}$ the class of all finite subdivisions $D=[I_{j},\hskip 0.85355ptj=1,\ldots,N]$ , $I_{j}=[a_{j-1},a_{j}]$ , $0=a_{0}<a_{1}<\cdots<a_{N}=1$ , and suppose that

\delta(D)\ =\ \max\left|I_{j}\right|\ =\ \max\{a_{j}-a_{j-1}\}.

Then

0\ <\ \delta(D)\ \leq\ 1

and the assumptions §9, #4 , are trivially satisfied. Suppose that

\begin{cases}\ \phi(I)=(\phi_{1},\ldots,\phi_{k})\\ \quad\text{where}\\ \ \phi_{r}=x_{r}(b)-x_{r}(a)\quad(r=1,\ldots,k)\end{cases}\forall\ I=[a,b]\subset A.

Denote by

\begin{cases}\ M_{r}=\max\limits_{0\leq t\leq 1}\left|x_{t}(t)\right|\\[4.0pt] \ M=\max\limits_{1\leq r\leq k}\ M_{r}\end{cases}.

Given $\varepsilon>0$ take $\eta=\eta(\varepsilon)=2$ . Let

D_{0}\ =\ [I_{i}:i=1,\ldots,N]

be any system $D_{0}\in\mathcal{D}$ (with $\delta(D_{0})<2)$ . Because of the continuity of $x(t)$ there is a $\lambda=\lambda(\varepsilon,D_{0})>0$ such that $\displaystyle\left|\phi(I)\right|<\frac{\varepsilon}{2N}$ for all $I=[a,b]$ with $b-a<\lambda$ . Now suppose $D=\left|J\right|$ is any system $D\in\mathcal{D}$ with $\delta(D)<\lambda$ . Then the sum $\sum{}^{\prime}\ \left\lVert\phi(J)\right\rVert$ contains at most $2(N-1)$ terms and

\sum{}^{\prime}\ \left\lVert\phi(J)\right\rVert\ <\ 2(N-1)\frac{\varepsilon}{2N}\ <\ \varepsilon.

For each $I_{i}\in D_{0}$ the intervals $J\in D$ , $J\subset I_{i}$ leave uncovered in $I$ at most two terminal intervals, say $H^{\prime}$ , $H^{\prime\prime}$ (if any). Then we have

\phi(I_{i})\ =\ \sum{}^{(i)}\ \phi(J)\hskip 1.42271pt+\hskip 1.42271pt\phi(H^{\prime})\hskip 1.42271pt+\hskip 1.42271pt\phi(H^{\prime\prime}),

	$\displaystyle\big{\|}\big{\|}\phi(I_{i})-\sum{}^{(i)}\ \phi(J)\big{\|}\big{\|}\$	$\displaystyle\leq\ \left\lVert\phi(H^{\prime})\right\rVert+\left\lVert\phi(H^{\prime\prime})\right\rVert$
		$\displaystyle<\ \frac{\varepsilon}{N},$

and

	$\displaystyle\sum\limits_{i=1,\ldots,N}\big{\|}\big{\|}\phi(I_{i})-\sum\limits_{\begin{subarray}{c}J\in D\\ J\subset I_{i}\end{subarray}}{}^{(i)}\ \phi(J)\big{\|}\big{\|}\$	$\displaystyle<\ N\hskip 0.85355pt\frac{\varepsilon}{N}$
		$\displaystyle=\ \varepsilon.$

Thus $\phi$ is quasi additive, as are $\phi_{r}$ , $\left\lVert\phi\right\rVert$ , $\phi_{r}^{+}$ , $\phi_{r}^{-}$ (cf. §9, #16). Obviously $V=V(\left\lVert\phi\right\rVert)$ is the Jordan length of the curve $C$ , and $V_{r}=V(\left|\phi_{r}\right|)$ , $V_{r}^{+}=V(\phi_{r}^{+})$ , $V_{r}^{-}=V(\phi_{r}^{-})$ , are the total variation, the positive and negative variations of $x_{r}(t)$ , $0\leq t\leq 1$ . And

V<+\infty\implies\begin{cases}\ \text{$C$ is rectifiable}\\[4.0pt] \ V_{r},\hskip 0.85355ptV_{r}^{+},\hskip 0.85355ptV_{r}^{-}\hskip 1.42271pt<\hskip 1.42271pt+\infty\\[4.0pt] \ \phi,\hskip 0.85355pt\phi_{r},\hskip 0.85355pt\left\lVert\phi\right\rVert,\hskip 0.85355pt\left|\phi\right|,\hskip 0.85355pt\phi_{r}^{+},\hskip 0.85355pt\phi_{r}^{-}\quad\text{are all quasi additive}\end{cases}.

myn8

EXAMPLE (Jordan length for discontinuous curves)

Let $C:x=x(t)$ , $0\leq t\leq 1$ , $x=(x_{1},\ldots,x_{k})$ , be any real vector function (a parameteric curve not necessarily continuous). Take $A=[0,1]$ , $\{I\}$ the class of all intervals $I=[a,b]\subset A$ , $\mathcal{D}$ the class of all finite subdivisions $D=[I_{j},\hskip 0.85355ptj=1,\ldots,N]$ , $I_{j}=[a_{j-1},a_{j}]$ , $0=a_{0}<a_{1}<\cdots<a_{N}=1$ . Let

s^{+}(t_{0})\ =\ \begin{cases}\limsup\limits_{t\downarrow t_{0}}\left\lVert x(t)-x(t_{0})\right\rVert\hskip 14.22636pt0\leq t_{0}<1\\[8.0pt] \quad 0\hskip 113.81102ptt_{0}=1\end{cases}.

Analogously, let

s^{-}(t_{0})\ =\ \begin{cases}\limsup\limits_{t\uparrow t_{0}}\left\lVert x(t)-x(t_{0})\right\rVert\hskip 14.22636pt0<t_{0}\leq 1\\[8.0pt] \quad 0\hskip 113.81102ptt_{0}=0\end{cases}.

Finally, let

s(t_{0})\ =\ s^{+}(t_{0})+s^{-}(t_{0})\qquad 0\leq t_{0}\leq 1.

As in the continuous situation, suppose that

\begin{cases}\ \phi(I)=(\phi_{1},\ldots,\phi_{k})\\ \quad\text{where}\\ \ \phi_{r}=x_{r}(b)-x_{r}(a)\quad(r=1,\ldots,k)\end{cases}\forall\ I=[a,b]\subset A.

Then

V\ =\ \sup\ \sum\ \left\lVert\phi(I)\right\rVert

is the Jordan length of $C$ , and $V<+\infty$ if and only if $x_{r}(t)$ , $r=1,\ldots,k$ are BV in $A$ . If $V<+\infty$ , then $s(t)$ , $0\leq t\leq 1$ , is zero everywhere but for countably many $t$ , and the sum (or the sum of the series)

\sigma\ =\ \sum\limits_{0\leq t\leq 1}\ s(t)\ \leq\ V

is finite. If we take

\delta(D)\ =\ \max\limits_{i=1,\ldots,N}\left|I_{i}\right|+\sigma-\sum\limits_{i=0}^{N}\ s(t_{i})

where

\begin{cases}\ D=[I_{1},\ldots,,I_{N}]\\ \ I_{i}=[a_{i-1},a_{i}]\quad 0=a_{0}<a_{1}<\cdots<a_{N}=1\end{cases},

then $\delta(D)$ is a mesh function and $\phi(I)$ , $I\in\{I\}$ is quasi additive w.r.t. $\delta(D)$ and $\mathcal{D}$ . We leave the proof to the reader. Then it is easy to prove also that

	$\displaystyle V\$	$\displaystyle=\ V\left(\left\lVert\phi\right\rVert\right)$
		$\displaystyle=\ \lim\limits_{\delta(D)\rightarrow 0}\ \sum\limits_{\begin{subarray}{c}I\in D\\ D\in\mathcal{D}\end{subarray}}\ \left\lVert\phi(I)\right\rVert.$

myn9

EXAMPLE (Cauchy integral in an interval in $\mathbb{R}^{m}$ )

We may suppose $A$ is the unit interval

A\ =\ \{(x_{1},\ldots,x_{m})\in\mathbb{R}^{m}:0\leq x_{r}\leq 1,r=1,\ldots,m\}.

Let $\mathcal{D}$ be the collection of all finite subdivisions $D=I$ of $A$ into intervals

I\ =\ [a_{r}\leq x_{r}\leq b_{r},r=1,\ldots,m].

Let $f(x),\ x\in A:\left|f(x)\right|\leq M$ be any bounded real function. For every $I$ , let $\begin{cases}\ \ell=\ell(I)\\ \ L=L(I)\end{cases}$ denote the $\begin{cases}\ \text{infimum}\\ \ \text{supremum}\end{cases}$ of $f(x)$ in $I$ , and let

\begin{cases}\ \Delta(I)=L-\ell\\ \ F=F(I),\quad F:\ell\leq F\leq L\\ \ \phi(I)=F(I)\left|I\right|,\quad\left|I\right|=(b_{1}-a_{1})\cdots(b_{m}-a_{m})\end{cases}.

Let

\delta(D)\ =\ \max\limits_{\begin{subarray}{c}I\in D,\\ D\in\mathcal{D}\end{subarray}}\ \text{diam}\hskip 1.42271ptI.

Then $\delta(D)$ is a mesh. The Riemann integrability condition reads:

(R) Given $\epsilon>0$ there is a $\sigma=\sigma(\epsilon)>0$ such that

\sum\ \Delta(I)\left|I\right|\ <\ \epsilon\quad\forall\ D\in\mathcal{D}\ \text{with}\ \delta(D)<\sigma.

It is easy to verify that (R) implies quasi additivity of $\phi(I)$ (w.r.t $\mathcal{D}$ and $\delta(D)$ ) and vice-versa. The limit

\mathcal{B}\ =\ \lim\ \sum\ \phi(I)

is the Cauchy integral of $f(x)$ in $A$ .

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EXAMPLE (Lebesgue-Stieltjes integral) Let $\mu$ be a measure in a $\sigma$ -ring $\mathcal{R}$ of subsets of a space $A$ , i.e., $(A,\mathcal{R},\mu)$ is a measure space, and let $f(x)$ , $x\in A$ , be a $\mu$ -measurable and $\mu$ -integrable function. It is not restrictive to suppose $f(x)\geq 0$ . For every $0<p<+\infty$ , and $0\leq p<q\leq+\infty$ , denote by $B(p)$ , $I(p,q)$ respectively the sets

B(p)\ =\ \{x\in A:f(x)=p\}

and

I(p,q)\ =\ \{x\in A:p<f(x)\leq q\}.

These sets are all $\mu$ -measurable and

\begin{cases}\ 0\leq p\hskip 0.85355pt\mu(B(p))<+\infty\\[4.0pt] \ \mu(B(+\infty))=0\end{cases},\ \text{and}\quad p\hskip 0.85355pt\mu(B(p))\equiv 0\quad\text{for}\quad p=+\infty.

And

\begin{cases}\ p\hskip 0.85355pt\hskip 0.85355pt\mu(I(p,q))<+\infty\\[11.0pt] \ \displaystyle\sum\limits_{p>0}\ p\hskip 0.85355pt\mu(B(p))\leq M<+\infty\quad(\exists\ M\geq 1)\hskip 28.45274pt\end{cases}.\hskip 64.01869pt

Thus the set $P$ of all $p>0$ with $\mu(B(p))>0$ is countable and the corresponding series

\sum\ p\hskip 0.85355pt\mu(B(p))

is convergent. Also

\begin{cases}I(p-\tau,p+\tau)\rightarrow B(p)\\[8.0pt] \mu(I(p-\tau,p+\tau))\rightarrow\mu(B(p))\end{cases}\tau\downarrow 0,\ \forall\ p>0.

Let $\{I\}$ be the collection of all sets

I(p,q)\quad 0\leq p<q\leq+\infty,

and let

\psi(I)\ =\ p\hskip 0.85355pt\mu(I(p,q)),\quad I\in\{I\}.

Let $\mathcal{D}$ be the family of all finite decompositions $D=[I_{i},\ i=1,\ldots,n]$ of $A$ into sets

I_{i}\ =\ \{x\in A:p_{i-1}<f(x)\leq p_{i}\}\in\{I\}

with

0=p_{0}<p_{1}<\cdots<p_{n-1}<p_{n}=+\infty.

Finally, let

\delta(D)\ =\ \begin{cases}\qquad 1\hskip 213.39566ptn=1\\[15.0pt] \displaystyle\ \max\limits_{i=1,\ldots,n=1}\ (p_{i}-p_{i-1})+\frac{1}{p_{n-1}}+\sum\limits_{i=1}^{n-1}\ p_{i}\hskip 0.85355pt\mu(B(p))\quad n>1\end{cases}.

Then $\delta(D)$ is a mesh function and $\psi(I)$ is quasi additive (w.r.t $\mathcal{D}$ and $\delta(D)$ ). To prove the last statement, given $\varepsilon>0$ , take

\eta\ =\ \eta(\varepsilon)\ =\ \min\hskip 1.42271pt\left[\frac{\varepsilon}{4M},1\right]

and let

D_{0}\ =\ [I_{i}=I(p_{i-1},p_{i}):i=1,\ldots,N],\quad 0=p_{0}<p_{1}<\cdots<p_{n-1}<p_{n}=+\infty

be any $D_{0}\in\mathcal{D}$ with $\delta(D_{0})<\eta$ . Then there are numbers $\tau_{i}>0$ such that

\mu(I(p_{i}-\tau_{i},p_{i}+\tau_{i})\ <\ \mu(B(p_{i}))+\frac{\varepsilon}{4\hskip 0.85355ptN\hskip 0.85355ptp_{i}},\ i=1,\ldots,N-1.

Take

\tau\ =\ \min\hskip 1.42271pt\tau_{i},

and

\lambda\ =\ \lambda(\varepsilon,D_{0})\ =\ \min\hskip 1.42271pt[p_{i}-p_{i-1},\hskip 1.42271pti=1,\ldots,N-1;\hskip 1.42271pt1/p_{N-1};\hskip 1.42271pt\tau;\hskip 1.42271pt\eta].

D\ =\ [J_{i}=(q_{j-1},q_{j}),j=1,\ldots,n],\quad 0=q_{0}<q_{1}<\cdots<q_{n-1}<q_{n}=+\infty,

is any $D\in\mathcal{D}$ with $\delta(D)<\lambda$ , we have $J_{1}\subset I_{1}$ , $J_{n}\subset I_{N}$ , and

	$\displaystyle\Delta\$	$\displaystyle=\ \sum\limits_{i=1}^{N}\ \left[\sum{}^{(i)}\ \psi(J_{i})-\psi(I_{i})\right]^{-}$
		$\displaystyle=\ \sum\limits_{i=1}^{N}\ p_{i-1}\hskip 0.85355pt\left(\sum{}^{(i)}\ \mu(J(q_{j-1},q_{j}))-\mu(I(p_{i-1},p_{i}))\right)^{-}$

where all $q_{j-1}$ , $q_{j}$ relative to $\sum{}^{(i)}$ are between $p_{i-1}$ and $p_{i}$ , and $q_{0}=p_{0}=0$ . We have

	$\displaystyle\Delta\$	$\displaystyle\leq\ \sum\limits_{i=2}^{N}\ p_{i-1}\hskip 0.85355pt\left(\sum{}^{(i)}\ \mu(J(q_{j-1},q_{j}))-\mu(I(p_{i-1},p_{i}))\right)^{-}$
		$\displaystyle\leq\ \sum\limits_{i=2}^{N}\ p_{i-1}\hskip 0.85355pt\left[\mu(B(p_{i-1}))+\frac{\varepsilon}{4}N\hskip 0.85355ptp_{i-1}+\mu(B(p_{i-1}))+\frac{\varepsilon}{4\hskip 0.85355ptN\hskip 0.85355ptp_{i}}\right]$
		$\displaystyle\hskip 85.35826pt+p_{N-1}\left[\mu(B(p_{N-1}))+\frac{\varepsilon}{4\hskip 0.85355ptN\hskip 0.85355ptp_{N-1}}\right]$
		$\displaystyle<\ 2\delta(D_{0})+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}$
		$\displaystyle<\ \frac{\varepsilon}{2}+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}$
		$\displaystyle=\ \varepsilon.$

Thus $\psi(I)$ , $I\in\{I\}$ , is quasi subadditive (w.r.t. $\mathcal{D}$ and $\delta(D)$ ), and hence

V\ =\ V(\psi)\ =\ \lim\limits_{\delta(D)\rightarrow 0}\ \sum\limits_{I\in D}\ \psi(I).

Since $f(x)$ is $\mu$ -integrable, we have

V\ =\ V(\psi)\ <\ +\infty,

and (cf. §9, #16), $\psi$ is quasi additive. Also it is easy to prove that

V(\psi)\ =\ (A)\ \int\ f(x)\ \text{d}\mu

is the Lebesgue-Stieltjes integral of $f(x)$ in $A$ (see, e.g. Rosenthal²²2Set Functions, Albuquerque, University of New Mexico Press, (1948). and Zaanen³³3Theory of Integration, New York, Interscience, (1958). ) It is also known that the limit above exists even if taken as

\lim\limits_{\Delta\rightarrow 0}\ \sum\limits_{I\in D}\ \psi(I)\qquad(\Delta\hskip 1.42271pt=\hskip 1.42271pt\max\ [p_{i}-p_{i-1}]+1/p_{n-1})

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EXAMPLE (Weierstrass integral over a rectifiable continuous curve $C$ )

Let $C:x=x(t)$ , $0\leq t\leq 1$ , $x=(x_{1},\ldots,x_{k})$ , be any real continuous vector function with finite Jordan length $L$ . As in Example #7, let $A=[0,1]$ , $I$ the class of all $I=[a,b]\subset A$ , $\mathcal{D}$ the class of all finite subdivisions $D=[I_{i}:i=1,\ldots N]$ , $I_{i}=[a_{i-1},a_{i}]$ , $0=a_{0}<a_{1}<\cdots<a_{N}=1$ , and

\delta\ =\ \delta(D)\ =\ \max\hskip 1.42271pt\{a_{i}-a_{i-1}\}.

And

\begin{cases}\ \phi(I)=(\phi_{1},\ldots,\phi_{k})\\ \quad\text{where}\\ \ \phi_{r}(I)=x_{r}(b)-x_{r}(a)\quad(r=1,\ldots,k)\end{cases}\forall\ I=[a,b]\subset A.

Then $L=V(\left\lVert\phi\right\rVert)$ (cf. Example #7) and the functions $\phi,\hskip 0.85355pt\phi_{r},\hskip 0.85355pt\left\lVert\phi\right\rVert,\hskip 0.85355pt\left|\phi\right|,\hskip 0.85355pt\phi_{r}^{+},\hskip 0.85355pt\phi_{r}^{-}$ are all quasi additive w.r.t. $D$ and $\delta(D)$ .

Let $K$ be any compact set containing the graph $[C]$ of $C$ , i.e.,

[C]\subset K\subset\mathbb{R}^{k},

and let $f(p,q)$ , $p\in K$ , $q\in\mathbb{R}^{k}$ , be any function continuous on $K\times\mathbb{R}^{k}$ such that

f(p,tq)\ =\ t\hskip 0.85355ptf(p,q),\quad\forall\ t\geq 0,\hskip 1.42271ptp\in K,\hskip 1.42271ptq\in\mathbb{R}^{k}.

For every $I\in\{I\}$ , let

\Phi(I)\ =\ f[x(\tau),\phi(I)]\qquad\tau\in I\quad\text{arbitary}.

Then (§10, #7) $\Phi(I)$ , $I\in\{I\}$ , is quasi additive w.r.t. $\mathcal{D}$ and $\delta(D)$ , and the numbers $\eta(\varepsilon)$ , $\lambda(\varepsilon,D_{0})$ can be taken independently of the choice of the $\tau$ 's in the intervals $I$ . Thus, the following limit exists and is finite:

	$\displaystyle\mathcal{F}\$	$\displaystyle=\ \int\limits_{C}\ f(p,q)$
		$\displaystyle=\ \lim\limits_{\delta(D)\rightarrow 0}\ \sum\limits_{I\in D}\ \Phi(I)$
		$\displaystyle=\ \lim\limits_{\delta(D)\rightarrow 0}\ \sum\limits_{i=1}^{n}\ f[x(\tau_{i}),x(a_{i})-x(a_{i-1})],$

independently of the choices of the points $\tau_{i}\in[a_{i-1},a_{i}]$ . This limit is known as the Weierstrass integral of $f$ on the rectifiable curve $C$ and was studied by L. Tonelli, G. Bouligand, N. Aronszajn, K. Menger, C. Y. Pauc. See Cesari²²2Surface Area, Annals of Mathematics Studies, Number 35, Princeton University Press, (1956). for references.

It is known that $\mathcal{F}$ is invariant w.r.t. Fréchet equivalence, i.e., it is independent of the representation $C:x=x(t)$ of $C$ . In particular, if $s=s(t)$ denotes the Jordan length of the curve $C_{t}$ defined by $x(t)$ on $[0,t]$ , $0\leq s\leq L$ , and $X(s)=x[s(t)]$ , then $C:x=x(s)$ , $0\leq s\leq L$ , is a representation of $C$ , $X(s)$ is Lipschitz of constant 1, and $\mathcal{F}$ is given by the Lebesgue integral

	$\displaystyle\mathcal{F}\$	$\displaystyle=\ \int\limits_{C}\ f(p,q)$
		$\displaystyle=\ \int\limits_{0}^{L}\ f[X(s),X^{\prime}(s)]\ \text{d}s.$

REFERENCES

BOOKS

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Cesari, Lamberto, Surface Area, Annals of Mathematics Studies, Number 35, Princeton University Press, 1956.

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Evans, L. C. and Gariepy, R. F., Measure theory and fine properties of functions, CRC Press, Boca Raton, 1992.

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Goffman, Casper, Nishiura, Toga, and Waterman, Daniel, Homeomorphisms in Analysis, American Mathematical Society, 1997.

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Goffman, C., Real Functions, Rinehart, New York, 1953.

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Leoni, Giovanni, A First Course in Sobolev Spaces, Second Edition, American Mathematical Society, Providence, Rhode Island, 2017. (pp. 133-155).

ARTICLES

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Burkill, J.C., Functions of Intervals, Proc. London Math. Soc. (2) vol. 22 (1924), 275-310.

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Cesari, L., Quasi-additive set functions and the concept of integral over a variety, Trans. Amer. Math. Soc. vol. 102 (1962), 94-113.

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Cesari, L., Variation, Multiplicity, and Semicontinuity, American Mathematical Monthly vol. 65, issue 5, (1958), 317-332.

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Cesari, L., Rectifiable Curves and the Weierstrass Integral, American Mathematical Monthly vol. 65, issue 7, (1958), 485-500.

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Fréchet, M., Sur l’integral d’une fonctionnelle etendue a un ensemble abstrait, Bull. Soc. Math. France. vol. 43 (1915), 249-267.

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Gavurin, M., Uber die Stieltjessche Integration abstrakter Funktionen, Fund. Math. vol. 27 (1936), 255-268.

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Goffman, C., Non-parametric surfaces given by linearly continuous functions, Acta Math. 103 (1960), 269-291.

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Goffman, C., A characterization of linearly continuous functions whose partial derivatives are measures, Acta Math. 117 (1967), 165-190.

{rf}

Henstock, R., On interval functions and their integrals, J. Lond. Math. Soc. vol. 21 (1946), 204-209.

{rf}

Kempisty, S., Fonctions d’intervalle non additives, Actualites Sci. Ind. No. 824, Paris Hermann, 1939.

{rf}

Kober, H., On the existence of the Burkill integral, Canad. J. Math. vol. 10 (1957), 115-121.

{rf}

Kolmogoroff, A., Untersuchungen uber den Intergralbergriff, Math. Ann. vol. 103 (1930), 654-696.

{rf}

Krickeberg, K., Distributionen, Funktionen beschrankter variation und Lebesguescher Inhalt nichtparametrischer Flachen, Ann. Mat. Pura Appl. 4 (1957), 105-133.

{rf}

McShane, E. J., Generalized Curves, Duke Math. Journal vol. 6 (1940), 513-536.

{rf}

Pollard, S., The stieltjes integral and its generalizations, Quart. J. Math. vol. 49 (1923), 73-138.

{rf}

Serrin, J., A new definition of the integral for non-parametric problems in the calculus of variations, Acta Math. vol. 102 (1959), 23-32.

{rf}

Smith, H. L., Stieltjes Integral, Trans. Amer. Math. Soc. vol. 27 (1925), 490-507.

{rf}

Warner, G., The Burkill-Cesari Integral, Duke Mathematical Journal vol. 35, #1, (1968), 61-78.

Ph. D. THESES

{rf}

Breckenridge, John Clark, Geocze K-Area and Measure Theoretical Methods in Surface Area Theory, 1969.

{rf}

Gariepy, Ronald Francis, Current Valued Measures and Geocze Area, 1969.

{rf}

Nishiura, T., Analytic Theory of Continuous Transformations, 1959.

{rf}

Shapley, L. S., Additive and Non-Additive Set Functions, 1953.

{rf}

Stoddart, A. W. J., Integrals of the Calculus of Variations, 1964.

{rf}

Turner, L. H., The Direct Method in the Calculus of Variations, 1957.

{rf}

Warner, Garth W., Quasi Additive Set Functions and Non-linear Integration over a Variety, 1966.

	$\displaystyle\bigg{\|}I(C)\hskip 1.42271pt-\hskip 1.42271pt\sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}),$	$\displaystyle\hskip 1.42271pt\underline{f}(x_{i})-\underline{f}(x_{i-1}))\bigg{\|}$
		$\displaystyle\leq\ \int\limits_{a}^{b}\ \left\|F(\underline{f}(x),\underline{\theta}(x))-g(x;P)\right\|\ \text{d}\mu_{C}$
		$\displaystyle=\ \int\limits_{[a,b]-S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}+\int\limits_{S_{P}}\ \left\|\ldots\right\|\ \text{d}\mu_{C}.$

	$\displaystyle\left\|I(C)-I(D)\right\|\$	$\displaystyle\leq\ \bigg{\|}I(C)-\sum\limits_{p=1}^{q}\ F(\underline{f}(\xi_{p}),\underline{f}(x_{p}^{})-\underline{f}(x_{p-1}^{}))\bigg{\|}$
		$\displaystyle\qquad+\sum\limits_{p=1}^{q}\ \left\|F(\underline{f}(\xi_{p}),\underline{f}(x_{p}^{})-\underline{f}(x_{p-1}^{}))-F(\underline{g}(\phi(\xi_{p})),\underline{g}(y_{p}^{})-\underline{g}(y_{p-1}^{}))\right\|$
		$\displaystyle\qquad\qquad+\bigg{\|}\sum\limits_{p=1}^{q}\ F(\underline{g}(\phi(\xi_{p})),\underline{g}(y_{p}^{})-\underline{g}(y_{p-1}^{}))-I(D)\bigg{\|}.$

	$\displaystyle\left\lVert\underline{f}(x_{p}^{})-\underline{f}(x_{p-1}^{})-\underline{g}(y_{p}^{})+\underline{g}(y_{p-1}^{})\right\rVert\$
	$\displaystyle\hskip 56.9055pt\leq\ \left\lVert\underline{f}(x_{p}^{})-\underline{g}(y_{p}^{})\right\rVert+\left\lVert\underline{f}(x_{p-1}^{})-\underline{g}(y_{p-1}^{})\right\rVert$
	$\displaystyle\hskip 56.9055pt=\ \left\lVert\underline{f}(x_{p}^{})-\underline{g}(\phi(x_{p}^{}))\right\rVert+\left\lVert\underline{f}(x_{p-1}^{})-\underline{g}(\phi(x_{p-1}^{}))\right\rVert$
	$\displaystyle\hskip 56.9055pt<\ \gamma+\gamma$
	$\displaystyle\hskip 56.9055pt=\ 2\gamma.$

	$\displaystyle\sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\$	$\displaystyle\Phi(I)-\Phi(I_{0})\Big{\|}\$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\ F(\zeta(I),\phi(I))-\sum\limits_{I\subset I_{0}}\ F(\zeta(I_{0}),\alpha(I_{0}))\left\lVert\phi(I)\right\rVert$
		$\displaystyle\qquad\qquad+\sum\limits_{I\subset I_{0}}\ F(\zeta(I_{0}),\alpha(I_{0}))\left\lVert\phi(I)\right\rVert-F(\zeta(I_{0}),\alpha(I_{0}))\left\lVert\phi(I_{0})\right\rVert\Big{\|}$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \Big{\|}\sum\limits_{I\subset I_{0}}\ F(\zeta(I),\alpha(I))-F(\zeta(I_{0}),\alpha(I_{0})))\left\lVert\phi(I)\right\rVert$
		$\displaystyle\qquad\qquad+\sum\limits_{I\subset I_{0}}\ F(\zeta(I_{0}),\alpha(I_{0}))\hskip 0.85355pt(\left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert)\Big{\|}$
		$\displaystyle\leq\ \sum\limits_{I_{0}}\Big{\|}F(\zeta(I_{0}),\alpha(I_{0}))\Big{\|}\quad\Big{\|}\sum\limits_{I\subset I_{0}}\left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{\|}$
		$\displaystyle\qquad\qquad+\sum\limits_{I_{0}}\ \sum\limits_{I\subset I_{0}}\ \|F(\zeta(I),\alpha(I))-F(\zeta(I_{0}),\alpha(I_{0}))\|\ \left\lVert\phi(I)\right\rVert$
		$\displaystyle=\ \sum\limits_{I_{0}}\ \Big{\|}F(\zeta(I_{0}),\alpha(I_{0}))\Big{\|}\quad\Big{\|}\sum\limits_{I\subset I_{0}}\ \left\lVert\phi(I)\right\rVert-\left\lVert\phi(I_{0})\right\rVert\Big{\|}$
		$\displaystyle\qquad+\sum\limits_{I_{0}}\ \Big{(}\sum{}_{\gamma^{-}}^{(I_{0})}+\sum{}_{\gamma^{+}}^{(I_{0})}\Big{)}\big{\|}F(\zeta(I_{0}),\alpha(I_{0}))-F(\zeta(I),\alpha(I))\big{\|}\left\lVert\phi(I)\right\rVert.$

	$\displaystyle\Big{\|}(\xi)\ \int\limits_{C}\ F\ \ -\ \$	$\displaystyle(\xi^{\prime})\ \int\limits_{C}\ F\ \Big{\|}\$
		$\displaystyle=\ \Big{\|}(\xi)\int\limits_{C}\ F-\sum\limits_{i=1}^{n}F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))$
		$\displaystyle\quad\qquad+\sum\limits_{i=1}^{n}F(\underline{f}(\xi_{i}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))$
		$\displaystyle\quad\qquad-\sum\limits_{i=1}^{n}F(\underline{f}(\xi_{i}^{\prime}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))$
		$\displaystyle\quad\qquad+\sum\limits_{i=1}^{n}\ F(\underline{f}(\xi_{i}^{\prime}),\underline{f}(x_{i})-\underline{f}(x_{i-1}))\ -\ (\xi^{\prime})\int\limits_{C}\ F\ \Big{\|}$

Analysis 101: Curves and Length

§\boldsymbol{\S}1. FUNDAMENTALS

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§\boldsymbol{\S}2. ESTIMATES

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§\boldsymbol{\S}3. EQUIVALENCES

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§\boldsymbol{\S}4. FRÉCHET DISTANCE

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§\boldsymbol{\S}5. THE REPRESENTATION THEOREM

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§\boldsymbol{\S}6. INDUCED MEASURES

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§\boldsymbol{\S}7. TWO THEOREMS

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Analysis 101:
Curves and Length

$\boldsymbol{\S}$ 1. FUNDAMENTALS

$\boldsymbol{\S}$ 2. ESTIMATES

$\boldsymbol{\S}$ 3. EQUIVALENCES

$\boldsymbol{\S}$ 4. FRÉCHET DISTANCE

$\boldsymbol{\S}$ 5. THE REPRESENTATION THEOREM

$\boldsymbol{\S}$ 6. INDUCED MEASURES

$\boldsymbol{\S}$ 7. TWO THEOREMS

$\boldsymbol{\S}$ 8. LINE INTEGRALS

$\boldsymbol{\S}$ 9. QUASI ADDITIVITY

$\boldsymbol{\S}$ 10. LINE INTEGRALS (bis)

$\boldsymbol{\S}$ 11. EXAMPLES