Microscopic and Strongly Microscopic Sets on the Plane. Fubini Theorem and Fubini Property

Abstract

Abstract. In this paper, we introduce the notions of microscopic and strongly microscopic sets on the plane and obtain a result analogous to Fubini Theorem.In measure theory or in functional analysis, it is often proved that some property holds "almost everywhere", i.e. except on some set of Lebesgue measure zero, or "nearly everywhere", that is except on some set of the first Baire category. Both these families, sets of Lebesgue measure zero and sets of the first category (on the real line, or generally in R n ), form σ-ideals, and moreover they are orthogonal to each other: there exist sets A and B such that R " A Y B, where A is a set of the first category and B is a nullset. Similarities and differences between these families are the main theme of the monograph "Measure and Category" of J. C. Oxtoby ([10]). The last part of these investigations is concerned with the Sierpiński-Erdös Duality Theorem. It leads to the Duality Principle, which allows us (assuming CH), in any proposition involving solely the notions of measure zero, first category, and notions of pure set theory, interchange the terms "nullset" and "set of the first category" whenever they appear. However, the extended principle, where the notions of measurability and the Baire property would be interchanged, is not true (see [10], Theorem 21.2 and Dual Statement).Fubini Theorem presents a close connection between the measure of any plane measurable set and the linear measure of its sections perpendicular to an axis. In [10] one can find an elementary proof of the fact that if E is a plane set of measure zero, then E x " ty : px, yq P Eu is a linear nullset for all x except a set of linear measure zero ([10], Theorem 14.2).2010 Mathematics Subject Classification: 28A75, 28A05. Key words and phrases: microscopic set, strongly microscopic set on the plane, Fubini Theorem, Fubini property.

Properties of the σ - ideal of microscopic sets

“…Hence we can obtain different notions of microscopic sets. The properties of the sets, their invariance with respect to translation, rotation and other algebraic and set-theoretic operations are investigated in [24].…”

Fubini Property for Microscopic Sets

“…The analogous considerations where carried out for microscopic sets by A. K a r a s iń s k a and E. W a g n e r-B o j a k o w s k a in [5].…”

“…In [5], it is proved, among other facts, that these families are σ-ideals of subsets of the plane situated between countable sets C 2 and sets of Lebesgue measure zero N 2 and essentially different from each of these families. Additionally, these σ-ideals are also orthogonal to the σ-ideal of sets of the first category on the plane.…”

“…In [5], the authors introduced the notion of microscopic and strongly microscopic sets on the plane. A set A ⊂ R 2 is microscopic if for each ε > 0 there exists a sequence {I n } n∈N of intervals, i.e., rectangles with sides parallel to coordinate axes, such that A ⊂ n∈N I n and m 2 (I n ) < ε n for each n ∈ N, where m 2 denotes two-dimensional Lebesgue measure.…”

“…In [5], it is proved, using the fact that the σ-ideals K and M are orthogonal, that neither the pair (K, M) nor (M, K) satisfies (FP). For this purpose, the authors used a microscopic set M of type G δ such that R \ M is a set of the first category (see also [4,Theorem 20.4…”