Maps of several variables of finite total variation and Helly-type selection principles

Details Maps of several variables of finite total variation and Helly-type selection principles Maps of several variables of finite total variation and Helly-type selection principles

2010-01-04

arxiv.org/abs/1001.0451v1

Journal of Mathematical Analysis and Applications, Vol. 370, No. 2 (2010), 672-686 (Part I), and Vol. 369, No. 1 (2010), 82-93

Mathematics

Functional Analysis

47 pages, LaTeX, uses elsarticle.sty

Scientific paper

Given a map from a rectangle in the n-dimensional real Euclidean space into a metric semigroup, we introduce a concept of the total variation, which generalizes a similar concept due to T. H. Hildebrandt (1963) for real functions of two variables and A. S. Leonov (1998) for real functions of n variables, and study its properties. We show that the total variation has many classical properties of Jordan's variation such as the additivity, generalized triangle inequality and sequential lower semicontinuity. We prove two variants of a pointwise selection principle of Helly-type, one of which is as follows: a pointwise precompact sequence of metric semigroup valued maps on the rectangle, whose total variations are uniformly bounded, admits a pointwise convergent subsequence.

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