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Regular version of the site

Совместный модуль вариации функций и условно регулярный принцип выбора

С. 399-402.
С.А.Чистякова, В.В.Чистяков

Given a closed interval $I=[a,b]$ and a metric space $(M,d)$, we introduce a

nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^I$ (the set of all

functions from $I$ into $M$), called the {\it joint modulus of variation}. We show that

if two sequences of functions $\{f_j\}$ and $\{g_j\}$ from $M^I$ are such that

$\{f_j\}$ is pointwise relatively compact on $I$, $\{g_j\}$ is pointwise convergent on $I$,

and $\limsup_{j\to\infty}\nu_n(f_j,g_j)=o(n)$ as $n\to\infty$, then $\{f_j\}$ admits

a pointwise convergent subsequence whose limit on $I$ is a conditionally regulated function.

In book

Т. 54: Теория функций, ее приложения и смежные вопросы. Каз.: Издательство Казанского математического общества и Академии наук РТ, 2017.