Special selectors of multivalued mappings. (Q1432510)

Comparing measurable and continuous selection theorems, the following natural question arises: assuming that a measurable set-valued mapping also satisfies the assumptions of \textit{E. Michael}'s [Ann. Math. (2) 63, 361--382 (1956; Zbl 0071.15902)] theorem on continuous selections on a subset \(\widetilde X\subseteq X\), does a measurable selection exist that is continuous at all points \(x\in\widetilde X\)? The author proves the following theorems giving an answer to this question: Theorem 1. Let \(B_0\simeq X\) be a closed set with a nonpositive topological dimension, and let \(B_1\subseteq X\) be such that the sets \(F(x)\) are convex for all \(x\in B_1\). Assume that \(F\) is lower semicontinuous at all points \(x\in B_0\cup\overline B_1\) (the overbar designates, as usual, the closure). Then, \(F\) has a measurable selection \(f\) that is continuous at all points \(x\in B=B_0\cup B_1\). Theorem 2. Let \(B_0\) be closed, its topological dimension be nonpositive, \(B_1\) be a measurable set, and \(F(x)\) be convex for all \(x\in B_1\). Assume that \(F\) is lower semicontinuous at all points \(x\in B_0\cup\overline B_1\). Then, \(F\) has a countable set of measurable selections \(\{f_i\}\) such that each function \(f_i\) is continuous at all points \(x\in B\), and the set \(\{f_i(x) \}^\infty_{i=1}\) is dense everywhere in \(F(x)\) for almost all \(x\).