A lower density operator for the Borel algebra

TL;DR

This paper establishes that the existence of a Borel lower density operator for countable sets in an uncountable Polish space is equivalent to the Continuum Hypothesis, linking set-theoretic axioms with measure-theoretic properties.

Contribution

It proves the equivalence between the existence of a Borel lower density operator and the Continuum Hypothesis in uncountable Polish spaces.

Findings

Existence of Borel lower density operator is equivalent to CH.

Provides a characterization linking measure theory and set theory.

Highlights the dependence of certain measure-theoretic constructs on set-theoretic axioms.

Abstract

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the $\sigma$-ideal of countable sets, for an uncountable Polish space, is equivalent to the Continuum Hypothesis.

Full text

Let $S$ be a $\sigma$-algebra of subsets of a nonempty set $X$ and let $J\subseteq S$ be a $\sigma$-ideal. We write $A\sim B$ whenever $A\bigtriangleup B\in J$. A mapping $\Phi\colon S\to S$ is called a lower density operator (respectively, a lifting) with respect to $J$ if it satisfies the following conditions (1)–(4) (respectively, (1)–(5)):

• (1)

$\Phi(X)=X$ and $\Phi(\emptyset)=\emptyset$,

• (2)

$A\sim B\implies\Phi(A)=\Phi(B)$ for every $A,B\in S$,

• (3)

$A\sim\Phi(A)$ for every $A\in S$,

• (4)

$\Phi(A\cap B)=\Phi(A)\cap\Phi(B)$ for every $A,B\in S$,

• (5)

$\Phi(A\cup B)=\Phi(A)\cup\Phi(B)$ for every $A,B\in S$.

The problem of the existence of liftings together with their various applications were widely discussed in the monograph [3] and in the later survey [9]. If $S$ is the $\sigma$-algebra of Borel sets in a given Hausdorff space, then the respective operator $\Phi$ satisfying conditions (1)–(5) is called a Borel lifting. Note that von Neumann and Stone [6] proved the existence of a lifting for a Borel measure space on $[0,1]$ under the assumption of the continuum hypothesis (CH). A simple proof of the same result was then given by Musiał [5]. This was later generalized by Mokobodzki [4] and Fremlin [1] who showed that, subject to CH, any $\sigma$-finite measure space with the measure algebra of cardinality $\leq\omega_{2}$ has a lifting. On the other hand, Shelah [7] proved that it is consistent with ZFC that there exists no Borel lifting for Lebesgue measure on $[0,1]$.

We will focus on a particular case. We assume that $S$ is the $\sigma$-algebra $\mathcal{B}$ of Borel subsets of an uncountable Polish space $X$ and $J$ is the $\sigma$-ideal $[X]^{\leq\omega}$ of all countable subsets of $X$. Since any two uncountable Borel subsets of Polish spaces are Borel isomorphic [8, Thm 3.3.13], it does not matter which Polish space is considered.

• (i)

CH*;*

• (ii)

there exists a lifting $\Phi\colon\mathcal{B}\to\mathcal{B}$ with respect to $[X]^{\leq\omega}$;

• (iii)

there exists a lower density operator $\Phi\colon\mathcal{B}\to\mathcal{B}$ with respect to $[X]^{\leq\omega}$.

To prove (iii)$\implies$(i) assume $\neg$CH. We work with $X:=\mathbb{R}\times\mathbb{R}$. Enumerate $\mathbb{R}$ as $\{x_{\alpha}\colon\alpha<\mathfrak{c}\}$. Suppose that $\Phi\colon\mathcal{B}\to\mathcal{B}$ is a lower density operator with respect to $[X]^{\leq\omega}$. Let $Q_{\alpha}:=\Phi(P_{\alpha})$ where $P_{\alpha}:=\{x_{\alpha}\}\times\mathbb{R}$ for $\alpha<\mathfrak{c}$. Note that if $\alpha\neq\beta$ then $Q_{\alpha}\cap Q_{\beta}=\Phi(P_{\alpha}\cap P_{\beta})=\emptyset$ by (4) and (1). Let $\pi_{2}\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be given by $\pi_{2}(x,y):=y$.

Claim. There is $x\in\mathbb{R}$ such that $\{\alpha<\mathfrak{c}\colon x\in\pi_{2}[Q_{\alpha}]\}$ is uncountable.

Proof of Claim. Suppose that $|\{\alpha<\mathfrak{c}\colon x\in\pi_{2}[Q_{\alpha}]\}|\leq\omega$ for each $x\in\mathbb{R}$. Let $L_{\alpha}:=\{\beta<\mathfrak{c}\colon x_{\alpha}\in\pi_{2}[Q_{\beta}]\}$ for $\alpha<\mathfrak{c}$. Then $|\bigcup_{\alpha<\omega_{1}}L_{\alpha}|\leq\omega_{1}$ by our supposition. By $\neg$CH, the set $\mathfrak{c}\setminus\bigcup_{\alpha<\omega_{1}}L_{\alpha}$ is nonempty (of cardinality $\mathfrak{c}$). Moreover, $\{x_{\alpha}\colon\alpha<\omega_{1}\}\cap\pi_{2}[Q_{\xi}]=\emptyset$ for each $\xi\in\mathfrak{c}\setminus\bigcup_{\alpha<\omega_{1}}L_{\alpha}$. Thus $\{x_{\alpha}\colon\alpha<\omega_{1}\}\subseteq\mathbb{R}\setminus\pi_{2}[Q_{\xi}]=\pi_{2}[P_{\xi}]\setminus\pi_{2}[Q_{\xi}]\subseteq\pi_{2}[P_{\xi}\setminus Q_{\xi}]$ which gives a contradiction since $|\pi_{2}[P_{\xi}\setminus Q_{\xi}]|\leq|P_{\xi}\setminus\Phi(P_{\xi})|\leq\omega$ by (3).

Take $x\in\mathbb{R}$ as in the Claim. Consider the closed set $P:=\mathbb{R}\times\{x\}$. Then $|P\cap P_{\alpha}|=1$ and $\Phi(P)\cap Q_{\alpha}=\Phi(P)\cap\Phi(P_{\alpha})=\Phi(P\cap P_{\alpha})=\emptyset$ for each $\alpha<\mathfrak{c}$, by (4), (2) and (1). Therefore $\Phi(P)\cap\bigcup_{\alpha<\mathfrak{c}}Q_{\alpha}=\emptyset$. On the other hand, $|P\cap\bigcup_{\alpha<\mathfrak{c}}Q_{\alpha}|>\omega$ by the choice of $x$. Thus $P\setminus\Phi(P)$ is uncountable and we reach a contradiction with (3). ∎

Note that implication (iii)$\implies$(ii) follows from [2, Theorem 2.8].

The above theorem answers a question posed by Jacek Hejduk during his invited talk given on the Conference on Real Function Theory in Stará Lesná in September 2016. He asked about the existence of a lower density operator on $\mathcal{B}(\mathbb{R})$ with respect to $[\mathbb{R}]^{\leq\omega}$. Let us mention that lower density operators play an important role in constructions of density like topologies; see [10], [2].

Acknowledgement. The first author would like to thank Kazimierz Musiał and Jacek Hejduk for their useful comments.