On strong algebrability of families of non-measurable functions of two variables

Let $\mathcal{L}$ be a vector space, $A\subseteq\mathcal{L}$ and $\kappa$ be a cardinal number. We say that $A$ is $\kappa$-lineable if $A\cup\{0\}$ contains a $\kappa$-dimensional subspace of $\mathcal{L}$. If we take $\mathcal{L}$ to be a commutative algebra, $A\subseteq\mathcal{L}$, then we say that $A$ is strongly $\kappa$-algebrable if $A\cup\{0\}$ contains a $\kappa$-generated subalgebra $B$ which is isomorphic to a free algebra.

Note that the set $X=\left\{x_{\alpha}\colon\alpha<\kappa\right\}$ is a set of free generators of some free algebra if and only if the set of all elements of the form $x_{\alpha_{1}}^{k_{1}}x_{\alpha_{2}}^{k_{2}}\cdots x_{\alpha_{n}}^{k_{n}}$, where $k_{1},k_{2},\dots,k_{n}$ are non-negative integers non-equal to $0$ and $\alpha_{1}<\alpha_{2}<\ldots<\alpha_{n}<\kappa$, is linearly independent; equivalently, for any $k\geq 1$, any non-zero polynomial $P$ in $k$ variables without a constant term and any distinct $x_{\alpha_{1}},\ldots,x_{\alpha_{k}}\in X$, we have that $P(x_{\alpha_{1}},\ldots,x_{\alpha_{k}})\in A\setminus\{0\}$. Note that if $P(x_{\alpha_{1}},\dots,x_{\alpha_{k}})$ is non-zero for any distinct $\alpha_{1},\dots,\alpha_{k}$, then $\{x_{\alpha}\colon\alpha<\kappa\}\subseteq A\setminus\{0\}$ (consider $P(x)=x$) and elements of $\{x_{\alpha}\colon\alpha<\kappa\}$ are different (consider $P(x,y)=x-y$). We will use this observation without mentioning it in every single proof of algebrability or lineability.

It turns out that $\mathbb{R}^{\mathbb{R}}$, or equivalently $\mathbb{R}^{\mathfrak{c}}$, contains a set of free generators with cardinality $2^{\mathfrak{c}}$, see [3].

Given a real function $f$ in one variable and a real function $F$ in two variables, we can define the Carathéodory superposition of $F$ and $f$ as a real function $F_{f}$ in one variable given by $F_{f}(x)=F(x,f(x))$. A function $F\colon\mathbb{R}^{2}\to\mathbb{R}$ is said to be sup-measurable if $F_{f}$ is Lebesgue measurable for every Lebesgue measurable $f\colon\mathbb{R}\to\mathbb{R}$. By [5, Lemma 3.4], it is sufficient to check the measurability of $F_{f}$ only for continuous functions $f$. There are measurable functions that are not sup-measurable: consider $F\colon\mathbb{R}^{2}\to\mathbb{R}$, $F=\chi_{X\times\{0\}}$, where $X\subseteq\mathbb{R}$ is non-measurable. The problem whether sup-measurable functions are measurable is undecidable in ZFC. On the one hand under the continuum hypothesis (CH) there is a sup-measurable function that is non-measurable, see [11] and [13] for the first such constructions. On the other hand there is a model of ZFC in which every sup-measurable function is measurable [16].

A function $F\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ is weakly sup-measurable if the superposition $F_{f}$ is measurable for any continuous and almost everywhere differentiable function $f\colon\mathbb{R}\rightarrow\mathbb{R}$.

For a function $F\colon\mathbb{R}^{2}\to\mathbb{R}$ and $y\in\mathbb{R}$ we denote by $F(\cdot,y)$ the horizontal section of $F$ at $y$, i.e. the function $x\mapsto F(x,y)$. Similarly, $F(y,\cdot)$ is the vertical section of $F$ at $y$.

We say that a function $F\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ is separately measurable if all horizontal and vertical sections of $F$ are measurable. A separately measurable function needs not to be measurable. To see this, consider a set $A\subseteq\mathbb{R}^{2}$ which has full outer measure but its intersection with each vertical and each horizontal line is a finite set (e.g. $A=\bigcup_{\alpha<\mathfrak{c}}A_{\alpha}$, where $A_{\alpha},\alpha<\mathfrak{c}$, are like in Lemma 1 for $Y=\mathbb{R}^{2}$). $A$ has full outer measure, but every vertical section of $A$ is null. Therefore, by Fubini’s Theorem, $A$ is non-measurable. Consider the characteristic function $\chi_{A}$ of $A$. Clearly $\chi_{A}$ is non-measurable. Let $x\in\mathbb{R}$. Then $\{y\in\mathbb{R}\colon(x,y)\in A\}$ has at most two elements. Since $y\mapsto\chi_{A}(x,y)$ takes non-zero values on a finite set, it is measurable. Similarly, $x\mapsto\chi_{A}(x,y)$ is measurable for every $y\in\mathbb{R}$. So $\chi_{A}$ is separately measurable. Note that if $F\colon\mathbb{R}^{2}\to\mathbb{R}$ has the property that $\{(x,y)\colon F(x,y)\neq 0\}\subseteq A$, then $F$ is separately measurable by the very same argument.

The following observation, which is a slight modification of [15, Lemma 11], will be a useful tool for us.

We say that that a function $f\colon\mathbb{R}\to\mathbb{R}$ is Darboux if it has the intermediate value property. We say that $f$ is Baire one if it is a pointwise limit of a sequence of continuous functions. The latter is equivalent to the fact that $f^{-1}[U]$ is $F_{\sigma}$ for any open $U\subseteq\mathbb{R}$.

The next simple lemma will be a useful tool in our investigations.

A function $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is called approximately continuous if it is continuous in the density topology, i.e. for any open set $U\subseteq\mathbb{R}$ the set $f^{-1}[U]$ is measurable and has density one at each of its points. It turns out that every approximately continuous function is Darboux of the first Baire class. If $N$ is a null set, then by [7, Theorem 6.5] there exists an approximately continuous function $g\colon\mathbb{R}\to[0,1]$ such that $g^{-1}(0)$ is a null cover of $N$. Then every $h_{n}=n\min\{g,1/n\}\colon\mathbb{R}\to[0,1]$ is approximately continuous as a composition of a continuous function $x\mapsto n\min\{x,1/n\}$ and an approximately continuous function $g$. Furthermore, $h_{n}^{-1}(0)$ is a null cover of $N$ and there exists $n$ such that the measure of $\mathbb{R}\setminus h_{n}^{-1}(1)$ is less than 1.

A function $f\colon\mathbb{R}\to\mathbb{R}$ is called a Jones function if for every closed set $K\subseteq\mathbb{R}^{2}$ with an uncountable projection on the $x$-axis we have $f\cap K\neq\emptyset$. Equivalently, if for any perfect subset $P\subseteq\mathbb{R}$ and continuous function $g\colon P\to\mathbb{R}$ we have $f\cap g\neq\emptyset$.

Jones functions were introduced by F. B. Jones in [12]. The author considered solutions of the Cauchy equation $f(x+y)=f(x)+f(y)$. He constructed a discontinuous function with a connected graph which satisfies the equation and the above definition.

One could ask if there is a function $F\colon\mathbb{R}^{2}\to\mathbb{R}$ for which all Carathéodory superpositions with continuous functions are Jones. Note that this problem is trivial – it suffices to define $F(x,y)=g(x)$, where $g$ is any Jones function. Therefore we also consider inverted Carathéodory superpositions: we say that a function $F\colon\mathbb{R}^{2}\to\mathbb{R}$ is sup-Jones if for every continuous function $f\colon\mathbb{R}\to\mathbb{R}$ the functions $F_{f}(x)\coloneqq F(x,f(x))$, $F^{f}(x)\coloneqq F(f(x),x)$ are Jones.

Consider the following condition.

1. (A)

   There exists a function $f\colon\mathbb{R}\to\mathbb{R}$ which is a union of a family of pairwise disjoint partial functions $\{f_{\alpha}\colon\alpha<\mathfrak{c}\}$ such that each $f_{\alpha}$ has positive outer measure and $\{x\in\mathbb{R}\colon f(x)=g(x)\}$ is a null set for each continuous $g\colon\mathbb{R}\to\mathbb{R}$.

In [17] von Weizsäcker noted that if $\operatorname{non}(\mathcal{N})\coloneqq\min\{|A|\colon A\subseteq\mathbb{R}\text{ is not a null set}\}=\mathfrak{c}$, then there exists a function $f\colon\mathbb{R}\to\mathbb{R}$ which has full outer measure and $\{x\in\mathbb{R}\colon f(x)=g(x)\}$ is a null set for every continuous $g\colon\mathbb{R}\to\mathbb{R}$. Under the same assumption, by [15, Lemma 8], such a function can be decomposed into $\mathfrak{c}$ many partial functions of full outer measure. In fact, with the assumption $\operatorname{non}(\mathcal{N})=\mathfrak{c}$ such a family of pairwise disjoint partial functions can be defined in a similar way to von Weizsäcker’s definition of a single function.

Later we will prove that condition (A) implies strong $2^{\mathfrak{c}}$-algebraility of the family of all sup-measurable functions which are non-measurable. Thus, by the result of Rosłanowski and Shelah, [16] condition (A) is independent of ZFC. It is unclear to us whether condition (A) is equivalent to the existence of a non-measurable sup-measurable function or not.

The minimal cardinal number $\kappa$ such that the real line can be covered by $\kappa$ many null sets is denoted by $\operatorname{cov}(\mathcal{N})$ and it is between $\omega_{1}$ and $\mathfrak{c}$. Similarly, the minimal cardinal number $\kappa$ such that some union of $\kappa$ many null sets is not null is denoted by $\operatorname{add}(\mathcal{N})$. Clearly $\omega_{1}\leq\operatorname{add}(\mathcal{N})\leq\operatorname{cov}(\mathcal{N})$. The equality $\operatorname{cov}(\mathcal{N})=\operatorname{add}(\mathcal{N})$ means that $\mathbb{R}$ can be covered by $\kappa$ many null sets but any union of less than $\kappa$ many of them is null, where $\kappa$ is the common cardinal $\operatorname{cov}(\mathcal{N})$ and $\operatorname{add}(\mathcal{N})$. This condition is independent of ZFC, see [4] for details. For example, it is fulfilled under CH, where both $\operatorname{cov}(\mathcal{N})$ and $\operatorname{add}(\mathcal{N})$ are $\omega_{1}$.

We say that $f\colon\mathbb{R}\to\mathbb{R}$ is almost perfectly everywhere surjective if its range $f[\mathbb{R}]$ is one of the following: $\mathbb{R},[0,\infty)$, or $(-\infty,0]$, and $f[P]=f[\mathbb{R}]$ for any perfect set $P\subset\mathbb{R}$. Note that this notion is different from that of a perfectly everywhere surjective function known in the literature, cf. [9]. Let us denote the family of all almost perfectly everywhere surjective functions by $\mathcal{APES}$. It follows from [3, Theorem 2.2] that $\mathcal{APES}$ is strongly $2^{\mathfrak{c}}$-algebrable.

This notion is connected to the following. A set $B\subseteq\mathbb{R}$ is called a Bernstein set if $B\cap P\neq\emptyset$ and $(\mathbb{R}\setminus B)\cap P\neq\emptyset$ for every perfect subset $P$ of $\mathbb{R}$. It is known that such sets are non-measurable. Let $f$ be an almost perfectly everywhere surjective function. Then $f^{-1}(x)$ is a Bernstein set for every $x\in f[\mathbb{R}]$. Therefore $f$ is non-measurable and $\{f^{-1}(x)\colon x\in f[\mathbb{R}]\}$ is a partition of $\mathbb{R}$.

A function $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is said to be approximately differentiable at a point $x\in\mathbb{R}$ if there exists a measurable set $E\subset\mathbb{R}$ such that $x$ is its density point, and the restriction $f\upharpoonright E$ is differentiable at $x$. A function $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is nowhere approximately differentiable if it is approximately differentiable at no $x\in\mathbb{R}$.

We will use the following observation. Let $g$ be a continuous and almost everywhere differentiable function, and let $f$ be a continuous and nowhere approximately differentiable function. Then the set $E\coloneqq\{x\in\mathbb{R}\colon f(x)=g(x)\}$ has measure zero. To see this, first note that $E$ is closed, since both $g$ and $f$ are continuous. Suppose, on the contrary, that $E$ has positive measure. By the Lebesgue Density Theorem, $E$ has a density point, say $x$, at which $g$ is differentiable. This implies that $f$ is approximately differentiable at $x$, which is a contradiction.

In [14] Lipiński constructed an example of a non-measurable function $F\colon\mathbb{R}^{2}\to\mathbb{R}$ whose all vertical and horizontal sections are Darboux Baire one functions. Another such construction is due to Grande [10, Theorem 2]. The paper is written in French and is therefore not easily accessible. Here we present Grande’s construction, slightly modified for our purposes. The difference is this. The function $h\colon X\to\left(0,1\right]$ used below was constant and equal to $1$ in Grande’s original construction.

Let $C\subseteq[0,1]$ be a Cantor set of positive measure with $0,1\in C$. Let $a_{0}=0$, $b_{0}=1$ and $\{(a_{n},b_{n})\colon n\geq 1\}$ be an enumeration of the gaps. Define

where $g_{n}\colon(a_{n},b_{n})\to(0,1]$, $n\geq 1$ are continuous surjections onto $(0,1]$, with

Due to the density of the gaps in $C$, any modification of the function $g$ on $C$ with values in $[0,1]$ preserves the intermediate value property. Let $B\subseteq C\setminus\bigcup_{n\geq 0}\{a_{n},b_{n}\}$ be a closed set with positive measure.

Let $X\subseteq B\times B$ be a non-measurable set in which all vertical and horizontal sections have at most one element (e.g. $X=\bigcup_{\alpha<\mathfrak{c}}A_{\alpha}$, where $A_{\alpha},\alpha<\mathfrak{c}$, are like in Lemma 1 for $Y=B\times B$). Let $h\colon X\to(0,1]$ be any function. Define

Note that $F_{h}^{-1}(0)\cap(C\times C)=(C\times C)\setminus X$, so $F_{h}$ is non-measurable.

We will show that all vertical and horizontal sections of $F$ are Darboux Baire one functions.

Let $x\in\mathbb{R}$. If $x\in\mathbb{R}\setminus B$, then $F_{h}(x,\cdot)$ is constant. Assume that $x\in B$. Then $F_{h}(x,\cdot)$ is either $g$ or $g$ modified at some point $y\in C$, where $(x,y)\in X$, so it has the intermediate value property. By Lemma 2, $F_{h}(x,\cdot)$ is also Baire one.

Let $y\in\mathbb{R}$. Consider the function

Note that $F_{h}(\cdot,y)$ is either $\ell$ or $\ell$ modified at one point $x\in C$, provided that $(x,y)\in X$, so it has the intermediate value property. By Lemma 2, $F_{h}(\cdot,y)$ is also Baire one.

We say that a function $f\colon\mathbb{R}\to\mathbb{R}$ is exponential like if

for some positive integer $n$, non-zero real numbers $\alpha_{1},\dots,\alpha_{n}$ and distinct non-zero real numbers $\beta_{1},\dots,\beta_{n}$. The notion was described in [2], where the Authors proved that if $\mathcal{A}\subseteq\mathbb{R}^{\mathbb{R}}$ is an arbitrary family and there exists a function $F\in\mathcal{A}$ such that $f\circ F\in\mathcal{A}$ for every exponential like function $f$, then $\mathcal{A}$ is strongly $\mathfrak{c}$-algebrable.

By an utrafilter on $\omega$ we mean any maximal non-trivial family of subsets of $\omega$ which is closed under taking supersets and finite intersections. Endowing $\omega$ with the discrete topology, we denote by $\beta\omega$ its Stone-Čech compactification, that is, the set of all ultrafilters on $\omega$ endowed with the topology which basic sets are of the form

where $a\subset\omega$. We will identify $\omega$ with the family of all principal ultrafilters $\delta_{n}=\{a\subset\omega\colon n\in a\}$, $n\in\omega$.

Using the fact that if $X$ is a compact space and $f\colon\omega\to X$ is a continuous function, then there exists a continuous extension $\overline{f}\colon\beta\omega\to X$ of $f$ (see e.g. [8]), we prove the following.