On functions of bounded variation

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 [20] 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.