1 A General Negative Result on Lower Densities

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 the symmetric difference \(A\bigtriangleup B\) is in J. This is an equivalence relation on S and its quotient space is denoted by S/J. A mapping \(\Phi :S\rightarrow S\) 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. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

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

  5. (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 [5] and in the later survey [14]. Lower density operators play an important role in constructions of density like topologies; see [4, 11, 15]. Several solutions of interesting problems on liftings require some set-theoretic tools. Our paper also use such methods.

We will start from a general negative result dealing with the negation of generalized continuum hypothesis (GCH). Then we will apply it to obtain a theorem stating the existence of a lower density operator on a Borel algebra in an uncountable Polish space, with respect to the \(\sigma \)-ideal of countable sets.

Let \(\kappa \) and \(\lambda \) be uncountable cardinals with \(\kappa \le \lambda \). Let \(|X|=\lambda \). A family F of subsets of X is called \(\kappa \)-additive if \(\bigcup G\in F\) whenever \(G\subseteq F\) and \(|G|<\kappa \). Note that an \(\omega _1\)-additive algebra is simply a \(\sigma \)-algebra. An algebra of subsets of \(X\times X\) is called a cross-type algebra if it contains all sets of the form \(\{ x\}\times X\) and \(X\times \{ x\}\) for \(x\in X\). Of course, all singletons \(\{(x,y)\}\), with \(x,y\in X\), belong to a cross-type algebra. Let \(J_{\kappa }\) denote the ideal of all subsets of \(X\times X\) of cardinality \(<\kappa \). Note that \(J_{\kappa }\) is contained in every \(\kappa \)-additive cross-type algebra of subsets of \(X\times X\).

Theorem 1

Assume that \(\kappa \) and \(\lambda \) are infinite cardinals such that \(\kappa ^+ <\lambda \). Fix a set X with \(|X|=\lambda \). Then for every \(\kappa ^+\)-additive cross-type algebra S of subsets of \(X\times X\), there is no lower density operator \(\Phi :S\rightarrow S\) with respect to the \(\sigma \)-ideal \(J_{\kappa ^+}\). In particular, this is true if \(\lambda :=2^\kappa \) and we assume that \(\kappa ^+ <2^\kappa \) (a part of \(\lnot \)GCH).

Proof

Enumerate X as \(\{x_\alpha :\alpha <\lambda \}\). Suppose that there exist \(\kappa ^+\)-additive cross-type algebra S of subsets of \(X\times X\) and a lower density operator \(\Phi :S\rightarrow S\) with respect to \(J_{\kappa ^+}\). Let \(Q_\alpha :=\Phi (P_\alpha )\) where \(P_\alpha :=\{x_\alpha \}\times X\) for \(\alpha <\lambda \). Note that if \(\alpha \ne \beta \) then \(Q_\alpha \cap Q_\beta =\Phi (P_\alpha \cap P_\beta )=\emptyset \) by (4) and (1). Let \(\pi _2:X\times X\rightarrow X\) be given by \(\pi _2(x,y):=y\). \(\square \)

Claim.There is\(x\in X\)such that\(|\{\beta <\lambda :x\in \pi _2[Q_\beta ]\}|\ge \kappa ^+\).

Proof of Claim

Suppose that \(\vert \{\beta<\lambda :x\in \pi _2[Q_\beta ]\}\vert <\kappa ^+\) for each \(x\in X\). Let

$$\begin{aligned} L_\alpha :=\{\beta<\lambda :x_\alpha \in \pi _2[Q_\beta ]\}\quad \text { for }\alpha <\lambda . \end{aligned}$$

Then \(\vert \bigcup _{\alpha <\kappa ^+}L_\alpha \vert \le \kappa ^+\) by our supposition. Since \(\kappa ^+<\lambda \), the set \(\lambda {\setminus }\bigcup _{\alpha <\kappa ^+}L_\alpha \) is nonempty (of cardinality \(\lambda \)). Take \(\xi \in \lambda {\setminus }\bigcup _{\alpha <\kappa ^+}L_\alpha \). Then

$$\begin{aligned} \{x_\alpha :\alpha <\kappa ^+\}\subseteq X{\setminus } \pi _2[Q_\xi ]=\pi _2[P_\xi ]{\setminus } \pi _2[Q_\xi ]\subseteq \pi _2[P_\xi {\setminus } Q_\xi ] \end{aligned}$$

which gives a contradiction since \(|\pi _2[P_\xi {\setminus } Q_\xi ]|\le |P_\xi {\setminus }\Phi (P_\xi )|<\kappa ^+\) by (3).

Take \(x\in X\) as in the Claim. Consider \(P:=X\times \{x\}\). 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 \(\alpha <\lambda \), by (4), (2) and (1). Therefore \(\Phi (P)\cap \bigcup _{\alpha <\lambda }Q_\alpha =\emptyset \). On the other hand, \(\vert P\cap \bigcup _{\alpha <\lambda }Q_\alpha \vert \ge \kappa ^+\) by the choice of x. Thus \(P{\setminus }\Phi (P)\notin J_{\kappa ^+}\) which yields a contradiction with (3). \(\square \)

2 A Theorem on the Existence of Borel Liftings

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 [9] 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 [8]. This was later generalized by Mokobodzki [6] and Fremlin [3] who showed that, subject to CH, any \(\sigma \)-finite measure space with the measure algebra of cardinality \(\le \omega _2\) has a lifting. On the other hand, Shelah [12] proved that \(2^\omega =\omega _2\) is consistent with the nonexistence of Borel lifting for the Lebesgue measure algebra. Later in [2], it was shown that \(2^\omega =\omega _2\) is consistent with the existence of Borel lifting for the Lebesgue measure algebra.

We focus on Borel liftings in the following case. We consider the \(\sigma \)-algebra \(\mathcal B(X)\) of Borel subsets of an uncountable Polish space X and the \(\sigma \)-ideal \([X]^{\le \omega }\) of all countable subsets of X. Since any two uncountable Borel subsets of Polish spaces are Borel isomorphic [13, Theorem 3.3.13], if we seek a lifting from \(\mathcal B(X)\) into \(\mathcal B(X)\) with respect to \([X]^{\le \omega }\), it does not matter which Polish space is considered.

Theorem 2

For an uncountable Polish space X, the following conditions are equivalent:

  1. (i)

    CH;

  2. (ii)

    there exists a lifting \(\Phi :\mathcal B(X)\rightarrow \mathcal B(X)\) with respect to \([X]^{\le \omega }\);

  3. (iii)

    there exists a lower density operator \(\Phi :\mathcal B(X)\rightarrow \mathcal B(X)\) with respect to \([X]^{\le \omega }\).

Proof

Implication (i)\(\implies \)(ii) follows from [8, Theorem 1] where it is shown that, for any \(\sigma \)-algebra S and any \(\sigma \)-ideal \(J\subseteq S\), if \(|S/J|\le \omega _1\), there exists a lifting from S to S, with respect to J.

Implication (ii)\(\implies \)(iii) is obvious.

To prove (iii)\(\implies \)(i) assume \(\lnot \)CH. We work with \(\mathbb R\times \mathbb R\) as a Polish space. It suffices to apply Theorem  1 where \(\kappa :=\omega \) and \(\lambda :=2^\omega =|\mathbb R|\). Then \(J_{\omega _1}\) consists of all countable subsets of the plane and the role of a cross-type \(\sigma \)-algebra is played by \(\mathcal B(\mathbb R\times \mathbb R)\). \(\square \)

Note that implication (iii)\(\implies \)(ii) follows from the final part of [11] or from [4, Theorem 2.8]. In fact, the existence of a lower density operator implies the existence of a lifting in a general case.

Theorem 2 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 \([\mathbb R]^{\le \omega }\).

3 Nonexistence of a Range of Bounded Borel Level

It can happen that the values of a lower density operator \(\Phi :S\rightarrow S\) with respect to \(J\subseteq S\) are located in a proper subfamily of S. Denote by \(\mathcal L\) the \(\sigma \)-algebra \(\mathcal L\) of Lebesgue measurable subsets of \(\mathbb R\). Recall a canonical lower density operator \(\Phi :\mathcal L\rightarrow \mathcal L\) with respect to the \(\sigma \)-ideal of null sets. Namely, \(\Phi \) is given by

$$\begin{aligned} \Phi (A):=\left\{ x\in \mathbb R:\lim _{h\rightarrow 0^+}\frac{\lambda (A\cap [x-h,x+h])}{2h}=1\right\} \end{aligned}$$

for \(A\in \mathcal L\) where \(\lambda \) stands for Lebesgue measure on \(\mathbb R\) (see [10, 15]). It is shown in [15] that the values of \(\Phi \) hit into the Borel class \({\pmb \Pi }^0_3\) (that is, \(F_{\sigma \delta }\) in the classic notation; cf. [13, 3.6]). In [1], an exact Borel complexity of sets \(\Phi (A)\) was studied where, instead of \(\mathbb R\), the Cantor space with the respective measure is considered.

In the above context, we return to lower density operators \(\Phi :\mathcal B(X)\rightarrow \mathcal B(X)\) which exist under CH by Theorem  2. We may ask whether the range of \(\Phi \) can be contained in \({\pmb \Sigma }^0_\alpha \) for some \(\alpha <\omega _1\) (we then say that this range is of bounded Borel level). We will show that the answer is negative. Pick \(\mathbb R\) as a Polish space and let \(\mathcal B:=\mathcal B(\mathbb R)\).

Proposition 3

There is no lower density operator \(\Phi :\mathcal B\rightarrow \mathcal B\) with respect to \([\mathbb R]^{\le \omega }\), whose range is of bounded Borel level.

Proof

Suppose that there exists \(\Phi \) with the range \(\Phi [\mathcal B]\subseteq {\pmb \Sigma }^0_\alpha \) for some \(\alpha <\omega _1\). We may assume that \(\alpha \ge 3\). Fix \(A\subseteq \mathbb R\) with \(A\in {\pmb \Pi }^0_\alpha {\setminus }{\pmb \Sigma ^0_\alpha }\) (cf. [13, Corollary 3.6.8]). We will show that \(\Phi (A)\notin {\pmb \Sigma }^0_\alpha \) which yields a contradiction. Suppose that \(\Phi (A)\in {\pmb \Sigma }^0_\alpha \). Since \(A\bigtriangleup \Phi (A)\) is countable, we have \(A=(\Phi (A)\cap B^c)\cup C\) for some countable sets \(B,C\subseteq \mathbb R\). Then \(B^c\in {\pmb \Pi }^0_2\subseteq {\pmb \Sigma }^0_\alpha \) and \(C\in {\pmb \Sigma }^0_2\subseteq {\pmb \Sigma }^0_\alpha \); see [13, Proposition 3.6.1]. Consequently, \(A\in {\pmb \Sigma }^0_\alpha \) which is impossible. \(\square \)

Finally, let us mention another negative result obtained in the recent paper [7]. Namely, it is proved that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference.