Selection theorems, based on generalized variation and oscillation
Theorems on the existence of convergent subsequences of a sequence of real functions, based on generalized variation and oscillation are proved. A notion of uniformly limited oscillation of real functions, a bit complicated to be reproduced here, is introduced. Among others the following theorem is proved for functions defined on a general set S. Let \(\{y_ i\}\) be a pointwise bounded sequence of real functions on a set S. If \(\{y_ k-y_{\ell}:\quad 1\leq \ell \leq k\}\) is of uniformly limited oscillation, then \(\{y_ i\}\) contains a subsequence which converges pointwise on S. In case the functions are defined on a subset of R, results closely connected with Helly's selection theorem are obtained. Also a new proof of Ascoli's theorem is given.
scientific article; zbMATH DE number 4061570