Subsequences and category (Q1970274)

Main result: Let \((X,\mu)\) be a regular locally compact \(\sigma\)-finite measure space, \(Z\) a separable Banach space, and \(f_n: X\to Z\) a sequence of Borel measurable functions. Suppose that the sequence \(f_n\) diverges \(\mu\)-a.e. Then \(\{s\in S|f_{s_n}\) diverges \(\mu\)-a.e.\} is a residual set in \(S\), where \(S\) stands for the set consisting of all increasing sequences of natural numbers.