On functions of bounded variation

TL;DR

This paper proves that functions with bounded Hardy--Krause variation are Borel measurable and have bounded $$-variation, and shows that this function space forms a commutative Banach algebra.

Contribution

It answers an open question about the measurability and boundedness of $$-variation functions and establishes a Banach algebra structure for this space.

Findings

Every function of bounded Hardy--Krause variation is Borel measurable.

Such functions have bounded $$-variation.

The space of bounded $$-variation functions forms a commutative Banach algebra.

Abstract

The recently introduced concept of $\mathcal{D}$-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from Pausinger \& Svane (J. Complexity, 2014) whether every function of bounded Hardy--Krause variation is Borel measurable and has bounded $\mathcal{D}$-variation. Moreover, we show that the space of functions of bounded $\mathcal{D}$-variation can be turned into a commutative Banach algebra.