On some problem of Sierpiński and Ruziewicz concerning the superposition of measurable functions. Microscopic Hamel basis.

Abstract

S. Ruziewicz and W. Sierpińnski in 1933 proved that each function

$f :\mathbb{R} \to\mathbb{R}$

can be
represented as a superposition of two measurable functions. Here the strengthening of
this theorem is given. The properties of Lusin set are studied and microscopic Hamel
bases are considered.