On some set-theoretical approaches to the question of measurability of functions

It is well known that set-theoretical 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 real-valued functions (in this context see, for instance, B.R. Gelbaum, J.M.H. Olmsted, Counterexamples in Analysis, Holden-Day, 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 sigma-finite 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.