Ukrainian mathematical congress  2009
Алекс Киртадзе (Тбилисский Математический Институт им. А. Размадзе, Грузия) On some settheoretical approaches to the question of measurability of functions It is well known that settheoretical methods play a significant role in various topics of modern real analysis and measure theory. For instance, the method of transfinite recursion, Bernstein`s classical construction of some pathological sets, Sierpinski`s functions with thick graphs, and others lead to many important results concerning measurability properties of realvalued functions (in this context see, for instance, B.R. Gelbaum, J.M.H. Olmsted, Counterexamples in Analysis, HoldenDay, San Francisco, 1964; A.B. Kharazishvili, Strange Functions in Real Analysis, Chapman and Hall/CRC, Second Edition, 2006). For two measurable spaces (E,S) and (E’,S’), which are equipped with sigmafinite measures m and m’ respectively, we present some sufficient conditions under which a function f acting from E into E’ has thick graph in the product space of E and E’ with respect to the product measure of m and m’. Such a function f is relatively measurable with respect to the class of all those measures which extend the original measure m (i.e., f becomes measurable with respect to a certain extension of m). On the other hand, some typical examples of absolutely nonmeasurable functions are presented by using a construction of Luzin type.
