The joint modulus of variation of metric space valued functions and pointwise selection principles

Abstract

Given T ⊂ R and a metric space M, we introduce a nondecreasing sequence of pseudometrics {ν n } on M T (the set of all functions from T into M), called the joint modulus of variation. We prove that if two sequences of functions {f j } and {g j } from M T are such that {f j } is pointwise precompact, {g j } is pointwise convergent, and the limit superior of ν n (f j , g j ) as j → ∞ is o(n) as n → ∞, then {f j } admits a pointwise convergent subsequence whose limit is a conditionally regulated function. We illustrate the sharpness of this result by examples (in particular, the assumption on the lim sup is necessary for uniformly convergent sequences {f j } and {g j }, and 'almost necessary' when they converge pointwise) and show that most of the known Helly-type pointwise selection theorems are its particular cases.

The Modulus of p-Variation and Its Applications

“…As a consequence, Chanturiya showed that estimates of Lebesgue and Oskolkov as well as Dini's criterion can be deduced from his more general estimate. The modulus of variation has found applications in other areas as well (see, e.g., [13,14,25]). The reader is referred to [1] and [16] for more information on moduli of variation.…”

The modulus of $p$-variation and its applications

“…As a consequence, Chanturiya showed that estimates of Lebesgue and Oskolkov as well as Dini's criterion can be deduced from his more general estimate. The modulus of variation has found applications in other areas as well (see, e.g., [26,14,13]). The reader is referred to [1] and [16] for more information on moduli of variation.…”