Pointwise selection theorems for metric space valued bivariate functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:452

Article

Chistyakov, Vyacheslav V.¹  Chistyakova, Svetlana A.¹ 

[1] Natl Res Univ Higher Sch Econ, Dept Informat Math & Comp Sci, Bolshaya Pecherskaya St 25-12, Nizhnii Novgorod 603155, Russia

关键词: Functions of several variables;    Metric space;    Total joint variation;    Pointwise convergence;    Selection principle;   

DOI  :  10.1016/j.jmaa.2017.03.053

来源: Elsevier

【 摘 要 】

We introduce a pseudometric TV on the set M-X of all functions mapping a rectangle X on the plane R-2 into a metric space M, called the total joint variation. We prove that if two sequences {f(j)} and {g(j)} of functions from M-X are such that {f(j)} is pointwise precompact on X, {g(j)} is pointwise convergent on X with the limit g is an element of M-X, and the limit superior of TV(f(j), g(j))as j -> infinity is finite, then a subsequence of {f(j)} converges pointwise on X to a function f is an element of M-X such that TV(f, g) is finite. One more pointwise selection theorem is given in terms of total epsilon-variations (epsilon > 0), which are approximations of the total variation as epsilon -> 0. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free