Some notes on monotone set-valued measures and Egoroff's theorem

In this note, we point out that Theorem 3 (a version of Egoroff's theorem for monotone set-valued measures) shown in the paper Lusin's theorem for monotone set-valued measures on topological spaces (Fuzzy Sets and Systems 364 (2019) 111-123) is not valid, and we present two revised versions. We also point out that the proofs of Lemma 2 and Theorem 1 are defective, thus the truth of their conclusions cannot be affirmed. Thereby, an open problem for the characteristics of monotone measures is raised. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

This note points out the invalidity of a theorem and the defects in two proofs in a paper, and raises an open problem regarding the characteristics of monotone measures.