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.