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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.

We investigate how large a set can be on which a continuous nowhere monotone function is one-to-one. We consider the σ-ideal of microscopic sets, which is situated between the countable sets and the sets of Hausdor dimension zero and prove that the typical function in C[0, 1] (in the sense of Baire)is nowhere monotone and one-to-one except on some microscopic set. We also give an example of a continuous nowhere monotone function of bounded variation on [0, 1], which is one-to-one except on some microscopic set, so it is not a typical function. The properties of continuous nowhere monotone functions were investigated by K. Padmavally [15], S. Marcus [13], K. M. Garg, A. M. Bruckner, C. E. Weil and others. K. M. Garg investigated level sets of such functions in a series of interesting papers [8], [9], [10], [11]. This problem was considered also by A. M. Bruckner and K. M. Garg in [6]. J. B. Brown, U. B.Darji and E. P. Larsen in [5] investigated the relationships between the continuous functions being monotone on no interval, monotone at no point, of monotonic type on no interval and of monotonic type at no point. Using two lemmas from [5] the rst named author proved in [12] that if f : [a, b] → R is a continuous nowhere monotone function then each set E having the Baire property such that f |E is one-to-one is of the rst category.

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

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