ABSTRACT
One of the earliest important results in real analysis and Lebesgue measure theory was obtained by Egorov, who discovered close relationships between the uniform convergence and the convergence almost everywhere of a sequence of real-valued Lebesgue measurable functions. This classical result (widely known now as Egorov’s theorem) has numerous consequences and applications in mathematical analysis and measure theory. This chapter discusses some aspects of Egorov’s theorem to show that, for a sequence of nonmeasurable real-valued functions, there is no hope of getting a reasonable analogue of this theorem. It demonstrates that there are some sequences of strange real-valued functions for which even weak analogues of Egorov type theorems fail to be true. In conformity with Egorov’s theorem, any convergent sequence of measurable real-valued functions converges uniformly on some large measurable subset of nonempty set E.
