Discrete stopping times in the lattice of continuous functions

We refer the reader to [AB03, AB06] for the theory of vector lattices. Given a Hausdorff topological space $K$, we will write $C(K)$ for the set of all continuous functions from $K$ to $\mathbb{R}$. The space $C(K)$ is a vector lattice under pointwise operations, and if $K$ is an extremally disconnected Hausdorff topological space, then $C(K)$ is order (or Dedekind) complete. A subset $A$ of a Hausdorff topological space is nowhere dense if $(\overline{A})^{\circ}=\emptyset$. This also implies that $\partial A$ is a nowhere dense set for every closed set $A$. A set $A$ is meagre if it is a union of a sequence of nowhere dense sets, and conversely, a set is co-meagre if its complement is meagre. $K$ is a Baire space if countable unions of closed sets with empty interior also have empty interior. Every compact Hausdorff space is a Baire space.

Given that $K$ is an extremally disconnected compact Hausdorff topological space, we write $C^{\infty}(K)$ for the vector lattice of all continuous functions from $K$ to $[-\infty,\infty]$ that are finite almost everywhere (a.e.), that is, except on a nowhere dense set. Two functions $f$ and $g$ in $C^{\infty}(K)$ are equal, provided that the set of points where their values differ is a nowhere dense set. Scalar multiplication, addition and lattice operations on $C^{\infty}(K)$ are defined pointwise a.e. It should be noted that $C^{\infty}(K)$ is an f-algebra with product defined pointwise a.e. Maeda-Ogasawara Theorem states that every Archimedean vector lattice $\mathcal{E}$ may be represented as an order dense sublattice of $C^{\infty}(K)$, where $K$ is an extremally disconnected compact Hausdorff topological space. Throughout the paper, we fix an order complete vector lattice $\mathcal{E}$ with a weak unit $E$ and a Maeda-Ogasawara representation of $\mathcal{E}$ as an order dense ideal in $C^{\infty}(K)$, where $K$ is the Stone space of $\mathcal{E}$, with $E$ corresponding to $\mathbbm{1}$.

We shall also use the concept of the sup-completion of $\mathcal{E}$ as introduced by [Don82] which we will denote by $\mathcal{E}^{s}$. We recall [[PT], Theorem 1] that states that the sup-completion of an order complete vector lattice $\mathcal{E}$ is $\mathcal{E}^{s}=\{f\in C(K,\overline{\mathbb{R}}):f\geq g\text{ for some }g\in\mathcal{E}\}$. We refer the reader to [PT] for more properties regarding the representation of $\mathcal{E}^{s}$. Every element of the sup-completion can be decomposed into finite and infinite parts [AN22, PT]. In terms of the above representation, this decomposition is as follows: the finite part $x$ of $u$ is defined as $x=P_{U}u$ where $U$ is the closure of $\{u<\infty\}$. The infinite part $w$ of $u$ is defined as $P_{U^{c}}u$ and is the function that vanishes on $U$ and takes the value $\infty$ on $U^{c}$. Corollary 7 in [Azo19, PT] also gives us that $\mathcal{E}_{+}^{u}\subseteq\mathcal{E}_{+}^{s}$.

The motivation for the following definitions and theory can be found in [KLW04a, KLW05]. A conditional expectation $\mathbb{F}$ defined on $\mathcal{E}$ is an order continuous strictly positive linear projection whose range $\mathcal{R}(\mathbb{F})$ is an order complete vector lattice of $\mathcal{E}$ with $\mathbb{F}E=E$. Conditional expectation operators satisfy an averaging property [[KLW05], Theorem 5.3] in the sense that if $f\in\mathcal{R}(\mathbb{F})$ and $g\in\mathcal{E}$ with $fg\in\mathcal{E}$ then $\mathbb{F}(fg)=f\mathbb{F}(g)$, where the product of two elements is done in $C^{\infty}(K)$. We note that the range of a conditional expectation operator is a regular sublattice of $\mathcal{E}$.

Let $J\subset\mathbb{R}_{+}$. A filtration on $\mathcal{E}$ is a family $(\mathbb{F}_{t})_{t\in J}$ of conditional expectations satisfying $\mathbb{F}_{s}=\mathbb{F}_{s}\mathbb{F}_{t}=\mathbb{F}_{t}\mathbb{F}_{s}$ for all $s\leq t$. We denote the range of $\mathbb{F}_{t}$ by $\mathcal{F}_{t}$. A stochastic process in $\mathcal{E}$ is a family $(X_{t})_{t\in J}$ where $X_{t}\in\mathcal{E}$, with $J\subset\mathbb{R}^{+}$ an interval. The stochastic process $(X_{t})_{t\in J}$ is right continuous if $\text{o-}\lim_{s\downarrow t}X_{s}=X_{t}$. The stochastic process $(X_{t})_{t\in J}$ is adapted to the filtration if $X_{t}\in\mathcal{F}_{t}$ for all $t\in J$. If $(X_{t})$ is a stochastic process adapted to $(\mathbb{F}_{t})$, we call $(X_{t},\mathbb{F}_{t})$ a super-martingale (respectively sub-martingale) if $\mathbb{F}_{t}(X_{s})\leq X_{t}$ (respectively $\mathbb{F}_{t}(X_{s})\geq X_{t}$) for all $t\leq s$. The process is called a martingale if it is a sub-martingale and super-martingale.

Denote by $F(\mathbb{R})$ the algebra consisting of all finite unions of disjoint left open right closed intervals $(a,b],(a,\infty)$ and $(\infty,b]$ with $a,b\in\mathbb{R}$. Let $\mathbb{L}$ be the vector lattice of real valued functions of the form:

where $(S_{i})_{i=1}^{n}$ is a partition of $\mathbb{R}$. The order relation of $\mathbb{L}$ is defined by $f\leq g$ if $f(t)\leq g(t)$ for every $t\in\mathbb{R}$.

We note that $I$ need not be an order continuous operator. Consider $I:\mathbb{L}\to\mathbb{R}$ where $I(f)=f(0)$. Then $I$ is a $\mathbb{R}-$valued Daniell integral and consider the sequence $f_{n}=\mathbbm{1}_{(-\frac{1}{n},0]}$. Then $f_{n}\downarrow 0$ in $\mathbb{L}$ but $I(f_{n})\not\to 0$.

We shall detail the construction of a specific Daniell integral developed by Grobler that shall be useful in defining a functional calculus on $\mathcal{E}$. For $Y\in\mathcal{E}$ we denote $P_{Y}$ to be the band projection associated with the band generated by $Y$. Fix $X\in\mathcal{E}$, then the right continuous spectral system of $X$ is the increasing right-continuous stochastic process $A=(A_{t})_{t\in\mathbb{R}}$ where $A_{t}=E-P_{(X-tE)^{+}}E$. We denote by $A_{\infty}$ and $A_{-\infty}$ respectively, the supremum and infimum of the process. To define the $\mathcal{E}-$valued Daniell integral, we define a vector lattice measure $\mu_{A}$ with respect to $(A_{t})_{t\in\mathbb{R}}$ as follows:

• •

  $\mu_{A}(a,b]=A_{b}-A_{a}$ where $(a,b]\in F(\mathbb{R})$.

• •

  For any finite disjoint union of half open intervals, we have $\mu_{A}(\cup_{i=1}^{n}I_{i})=\sum_{i=1}^{n}\mu_{A}(I_{i})$

Then $\mu_{A}$ is a countably additive $\mathcal{E}-$valued measure on $F(\mathbb{R})$ as shown in [Lemma 3.7, [Gro14b]] and the functional calculus is defined for elements of $\mathbb{L}$ as:

Define $\mathbb{L}^{\uparrow}:=\{f:\mathbb{R}\to\mathbb{\overline{R}}:\exists(f_{n})_{n\in\mathbb{N}}\subset\mathbb{L},\text{ such that }f_{n}(t)\uparrow f(t)\text{ for every }t\in\mathbb{R}\}$. Then the integral $I:\mathbb{L}\to\mathcal{E}$ can be extended to an integral from $\mathbb{L}^{\uparrow}$ to $\mathcal{E}^{s}$ as follows: for $f\in\mathbb{L}^{\uparrow}$, we define $I(f)=\sup_{n}I(f_{n})$ where $(f_{n})$ is a sequence in $\mathbb{L}$ such that $f_{n}\uparrow f$. Then [Lemma 3.2, [Gro14b]] states that this extension is well-defined. The extension also satisfies the following properties.

For $A\subseteq\mathbb{\overline{R}}$ and $\tau:K\to\overline{\mathbb{R}}$, we shall use the notation $\{\tau\in A\}$ to denote the set $\{\omega\in K:\tau(\omega)\in A\}$. We will need the following criterion for the convergence of sequences in a sup-completion. This essentially follows from the results in Section 3 of [BT22]. However they are not stated there in the form of the sup-completion, and thus we repeat certain arguments in the next two lemmas.

The above topological property allows us to deduce properties regarding the convergence in sup-completion.

Given a continuous function $f\in C(\mathbb{R})$ and $X\in C^{\infty}(K)$, let $U$ be the set on which $X$ is finite. Clearly, $U$ is open and dense, and the composition of $f$ and $X$ is defined, finite, and continuous on $U$. Then the composition extends uniquely to a function in $C^{\infty}(K)$. We shall denote this extended function by $f\circ X$. Also note that by the Maeda-Ogasawara theory, there is a one-to-one correspondence between clopen subsets of $K$ and bands in $\mathcal{E}$. For an element $Y\in\mathcal{E}$, the clopen set corresponding to the band generated by $Y$ is the set $\overline{\{Y\neq 0\}}$. Hence the corresponding band projection of $E$ is $P_{Y}E=\mathbbm{1}_{\overline{\{Y\neq 0\}}}$.

In view of the above theorem, we will show that for continuous functions the Daniell functional calculus satisfies a nice representation. The idea for the proof of the theorem can be found in Groblers’ paper. Let $\mathbb{L}^{\uparrow}_{0}$ be the set of all real valued positive functions in $\mathbb{L}^{\uparrow}$ and

Then analogous to the proof of [Proposition 3.5, [Gro14b]] we can show that $\mathbb{L}_{u}$ is a vector space and $I$ has a well-defined extension to $\mathbb{L}_{u}$ by defining for $f=g-h\in\mathbb{L}_{u}$, $I(f)=I(g)-I(h)$. The proof also shows that the extension is positive and linear on $\mathbb{L}_{u}$.

The following are some simple corollaries resulting from the above theorem.

The following proposition is a stronger statement than [Proposition 4.6, [Gro14b]] and [Proposition 2.8, [AT17]]. Recall the following convergence criterion for elements of $C^{\infty}(K)$ from [Theorem 3.7, [BT22]] that states that $x_{n}\xrightarrow{\mathrm{uo}}x$ if and only if $x_{n}$ converges to $x$ pointwise on a co-meagre set.

Since the Daniell functional calculus for continuous functions corresponds to the pointwise composition of functions, we can extend this to the concept of multivariate continuous functions. Given $f\in C(\mathbb{R}^{n},\mathbb{R})$ and $\mathbf{X}=(X_{i})_{i=1}^{n}\subset\mathcal{E}$ where $n\in\mathbb{N}$, let $U$ be the set on which all the $X_{i}$ are finite. Then $U$ is an open dense set and $f(X_{1},\dots,X_{n})$ is a well-defined continuous function on $U$. Then denote $f(\mathbf{X})$ to be the unique extension of $f(X_{1},\dots,X_{n})$ in $C^{\infty}(K)$. This enables us to prove the multivariate version of Jensen’s inequality. The univariate version of the Jensen’s inequality in the setting of vector lattices was proved in [Theorem 4.4, [Gro14b]]. Let $\mathbf{X}=(X_{1},\dots,X_{n})$ and for a given conditional expectation operator $\mathbb{F}$ on $\mathcal{E}$ let $\mathbb{F}\mathbf{X}:=(\mathbb{F}X_{1},\dots,\mathbb{F}X_{n})$.