Selection principle for pointwise bounded sequences of functions

Tret’yachenko, Yu. V.; Chistyakov, Vyacheslav V.

doi:10.1134/s0001434608090101

Cited by 10 publications

(1 citation statement)

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“…Note that the limit of a pointwise convergent subsequence of {f j } in Theorem 5 may be a non-regulated function. For more details, examples and relations with previously known 'regular' and 'irregular' versions of pointwise selection principles from [23,40,42] we refer to [14, Section 5.2], [17,18,34].…”

Section: Extensions Of Known Selection Theoremsmentioning

confidence: 99%

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

Chistyakov

Chistyakova

2017

Studia Math.

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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.

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