Measure-category properties of Borel plane sets and Borel functions of two variables

Abstract

Let I ⊗ J stand for the Fubini-type product of σ-ideals I, J ⊂ P(R). We consider mixed measure-category σ-ideals M ⊗ N and N ⊗ M (called the Mendez σ-ideals), and study some features of their structure. We show that M ⊗ N and N ⊗ M are not invariant with respect to nonzero rotations. Using Fremlin's results, we describe nice Borel bases of M ⊗ N, N ⊗ M, {∅} ⊗ N and {∅} ⊗ M. The rest of the paper is devoted to uniform versions of the Nikodym theorem and the Lusin theorem for Borel functions of two variables. Some σ-ideals on the planeThe monograph of Oxtoby [15] presents several similarities and differences between two families of small subsets of the real line that form σ-ideals: the family N of Lebesgue null sets (i.e. sets of Lebesgue measure zero) and the family M of meager sets (i.e. sets of the first Baire category). Oxtoby continues these studies for the σ-algebras associated with N and M (they consist of Lebesgue measurable sets and of sets with the Baire property, respectively) and real-valued functions measurable with respect to these σ-algebras. The analogous investigations can be conducted for subsets of R 2 or R k with k > 2. Interesting properties arise when one examines small sections of small plane sets -this leads to the theorems of Fubini and of Kuratowski and Ulam.A new idea appears when one considers plane sets whose almost all sections in one sense are small. Families (in fact σ-ideals) of such plane sets fulfilling the "Fubini-type mixed condition" were investigated by Mendez [13], [14]. These objects have some interesting properties and applications; see e.g. [6], [7], [8], [9], [2], [3], [4]. We show that detailed studies of the structure of the Mendez σ-ideals yield new properties of plane Borel sets and Key words and phrases: Lebesgue measure, Baire category, Mendez σ-ideal, nice base of a σ-ideal, Nikodym's theorem, Lusin's theorem.

On monotone hull operations

“…Note that these σ-ideals are mutually incomparable (with respect to inclusion) and also incomparable with M ⊗ M and N ⊗ N (see, e.g., [1] It is known that, if a σ-ideal I has a base of Borel sets and contains all singletons {x}, x ∈ R n , then H(B I) = I (we use the fact that a Bernstein nonmeasurable set is not in B I, cf. e.g., [9]).…”

“…Consequently, ϕ is additive. To prove that I ⊆H ,I ∈I ϕ (I) \ H ∈ I for all H ∈ H, it suffices (as in [5]) to show that for any I ∈ I and α < ω 1 …”

Ideals with bases of unbounded Borel complexity

“…It is well‐known that every Borel function on [0, 1] 2 is nice on a big domain (of Baire class 1 on a set of Lebesgue measure 1, continuous on a co‐meager set). However, in 3 it is proved that this is not the case if instead of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {Null}$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {Meager}$\end{document} ideals we would consider \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}(\mathsf {Null})$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}(\mathsf {Meager})$\end{document}. More precisely, for every α ≥ 2 there is a Borel function g : [0, 1] 2 → [0, 1] such that for every Borel set \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$M \in \mathcal {M}(\mathsf {Null})$\end{document} (or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}(\mathsf {Meager}))$\end{document} we can find x ∈ [0, 1] such that the function g x | M c x is not \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$ \Sigma ^0_\alpha$\end{document}‐measurable (here g x ( y ) = g ( x , y )).…”

“…Notice that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}(\mathcal {I})$\end{document} can be seen as \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \emptyset \rbrace \otimes \mathcal {I}$\end{document}. Fubini products of null or meager ideals have bases of bounded Borel complexity (e.g., \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {Null}\otimes \mathsf {Null}$\end{document}, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {Null}\otimes \mathsf {Meager}$\end{document}, cf., e.g., 3), so we cannot replace {∅︁} by \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {Null}$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {Meager}$\end{document} if we want to obtain a σ‐ideal with the complex Borel base property. However, we shall show that a σ‐ideal of the form \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {J}\otimes \mathsf {Null}$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(\mathcal {J}\otimes \mathsf {Meager})$\end{document} has the complex Borel base property if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {J}$\end{document} has property (M) and a Borel base.…”

“…On the other hand (see 3), these pathological points can be covered by null and meager sets, i.e., for every M as above, the set …”