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

TL;DR

This paper introduces a generalized concept of total variation for maps into metric semigroups, extending classical ideas and establishing Helly-type selection principles for sequences with bounded variation.

Contribution

It generalizes the notion of total variation for functions into metric semigroups and proves Helly-type selection principles in this broader context.

Findings

Total variation retains classical properties like additivity and lower semicontinuity.

Sequences with uniformly bounded total variation have pointwise convergent subsequences.

The concept extends classical variation theories to metric semigroup-valued maps.

Abstract

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.