Properties of the σ - ideal of microscopic sets

Abstract

The investigation of σ -ideals of subsets of the real line has a long tradition. The main motivation for the study of the collection of all microscopic sets, which constitutes a σ -ideal, stems from the fact that whenever one has to prove that a certain property in functional analysis or measure theory is fulfilled for "almost all" elements, the concept of "smallness" of the set of "exceptional points" should be described. The most classical of these concepts are related to Lebesgue nullsets and the sets of first Baire category.In certain applications, the ideals of measure and category turn out not to be suitable. In these situations it is useful to consider another σ -ideal having some good set-theoretic, algebraic and geometric properties.What makes microscopic sets interesting is the property that the collection of all microscopic sets constitutes a σ -ideal strictly smaller then the σ -ideal of sets of Lebesgue measure zero and orthogonal to the σ -ideal of sets of first Baire category. Therefore, in cases where it is well-known that a certain property holds everywhere except for a set of Lebesgue measure zero, it is important to check if the set of exceptional points is microscopic. If the answer is positive we get a stronger version of the property in question. In the classi-http://dx.doi.org/10.18778/7525-971-1.20

Resolvability of Measurable Spaces

“…where M denotes the σ-ideal of microscopic sets, is not topological and it is not weakly resolvable. For the definition and properties of microscopic sets, see [10].…”

On some generic small Cantor spaces

“…Microscopic subsets of the real line were introduced in [1]. Since then, they have been widely studied (see, for instance, [2], [3], [13], [12], [14], [18], [15], [16], [17], [19], [11], [21], [22]). The collection of microscopic subsets of the real line is a σ-ideal and it is a proper subset of the family of the Hausdorff zero dimensional sets ( [2], [14]).…”

“…Since then, they have been widely studied (see, for instance, [2], [3], [13], [12], [14], [18], [15], [16], [17], [19], [11], [21], [22]). The collection of microscopic subsets of the real line is a σ-ideal and it is a proper subset of the family of the Hausdorff zero dimensional sets ( [2], [14]). In [3], the authors consider the family CS of symmetric Cantor subsets of [0, 1], and among other results, they obtain properties concerning the subfamily of microscopic sets.…”