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 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.