A Lower Density Operator for the Borel Algebra

We prove that the existence of a Borel lower density operator (a Borel lifting) with respect to the σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-ideal of countable sets, for an uncountable Polish space, is equivalent to CH. One of the implications is known (due to K. Musiał) and the remaining implication is derived from a general abstract result dealing with the negation of GCH. We observe that there is no lower density Borel operator with respect to the σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-ideal of countable sets, whose range is of bounded level in the Borel hierarchy.


A General Negative Result on Lower Densities
Let S be a σ-algebra of subsets of a nonempty set X and let J ⊆ S be a σ-ideal. We write A ∼ B whenever the symmetric difference A B is in J. This is an equivalence relation on S and its quotient space is denoted by S/J. A mapping Φ : S → 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) Φ(X) = X and Φ(∅) = ∅, 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 σ-ideal of countable sets.
Note that an ω 1 -additive algebra is simply a σ-algebra. An algebra of subsets of X × X is called a cross-type algebra if it contains all sets of the form {x} × X and X × {x} for x ∈ X. Of course, all singletons {(x, y)}, with x, y ∈ X, belong to a cross-type algebra. Let J κ denote the ideal of all subsets of X × X of cardinality < κ. Note that J κ is contained in every κ-additive cross-type algebra of subsets of X × X. Theorem 1. Assume that κ and λ are infinite cardinals such that κ + < λ. Fix a set X with |X| = λ. Then for every κ + -additive cross-type algebra S of subsets of X × X, there is no lower density operator Φ: S → S with respect to the σ-ideal J κ + . In particular, this is true if λ := 2 κ and we assume that κ + < 2 κ (a part of ¬GCH).

A Theorem on the Existence of Borel Liftings
If S is the σ-algebra of Borel sets in a given Hausdorff space, then the respective operator Φ 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 Musia l [8]. This was later generalized by Mokobodzki [6] and Fremlin [3] who showed that, subject to CH, any σ-finite measure space with the measure algebra of cardinality ≤ ω 2 has a lifting. On the other hand, Shelah [12] proved that 2 ω = ω 2 is consistent with the nonexistence of Borel lifting for the Lebesgue measure algebra. Later in [2], it was shown that 2 ω = ω 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 σ-algebra B(X) of Borel subsets of an uncountable Polish space X and the σ-ideal [X] ≤ω 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 B(X) into B(X) with respect to [X] ≤ω , it does not matter which Polish space is considered. Proof. Implication (i) =⇒ (ii) follows from [8,Theorem 1] where it is shown that, for any σ-algebra S and any σ-ideal J ⊆ S, if |S/J| ≤ ω 1 , there exists a lifting from S to S, with respect to J. Implication (ii) =⇒ (iii) is obvious.
To prove (iii) =⇒ (i) assume ¬CH. We work with R × R as a Polish space. It suffices to apply Theorem 1 where κ := ω and λ := 2 ω = |R|. Then J ω1 consists of all countable subsets of the plane and the role of a cross-type σ-algebra is played by B(R × R).
Note that implication (iii) =⇒ (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.

Nonexistence of a Range of Bounded Borel Level
It can happen that the values of a lower density operator Φ : S → S with respect to J ⊆ S are located in a proper subfamily of S. Denote by L the σ-algebra L of Lebesgue measurable subsets of R. Recall a canonical lower density operator Φ : L → L with respect to the σ-ideal of null sets. Namely, Φ is given by for A ∈ L where λ stands for Lebesgue measure on R (see [10,15]). It is shown in [15] that the values of Φ hit into the Borel class Π Π Π 0 3 (that is, F σδ in the classic notation; cf. [13, 3.6]). In [1], an exact Borel complexity of sets Φ(A) was studied where, instead of R, the Cantor space with the respective measure is considered.
In the above context, we return to lower density operators Φ : B(X) → B(X) which exist under CH by Theorem 2. We may ask whether the range of Φ can be contained in Σ Σ Σ 0 α for some α < ω 1 (we then say that this range is of bounded Borel level). We will show that the answer is negative. Pick R as a Polish space and let B := B(R).

Proposition 3.
There is no lower density operator Φ: B → B with respect to [R] ≤ω , whose range is of bounded Borel level.
Proof. Suppose that there exists Φ with the range Φ[B] ⊆ Σ Σ Σ 0 α for some α < ω 1 . We may assume that α ≥ 3. Fix A ⊆ R with A ∈ Π Π Π 0 α \Σ Σ Σ 0 α (cf. [13, Corollary 3.6.8]). We will show that Φ(A) / ∈ Σ Σ Σ 0 α which yields a contradiction. Suppose 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.
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