On the transformations of measurable sets and sets with the Baire property (Q1890613)

Let \(\{f_n\}\) be a sequence of continuous strictly increasing functions defined on the unit interval and uniformly convergent to the identity function. Then for each set \(A\subset [0, 1]\) having the Baire property the sequence of characteristic functions of the sets \(f_n(A)\) converges to a characteristic function of the set \(A\) except on a set of the first category. If the set \(A\) is Lebesgue measurable, then the behaviour of the sequence of characteristic functions is more complicated and can be described in terms of properties of functions \(g_n\) and \(h_n\) from the Lebesgue decomposition of \(f_n\). The results are related to the paper by the second author and \textit{A. Kharazishvili} [Soobshch. Akad. Nauk Gruz. 145, No. 1, 43-46 (1992; Zbl 0770.28002)], where \(f_n(x)= x+ a_n\), \(\lim a_n= 0\).