On metric semigroups-valued functions of bounded Riesz-$$\Phi $$Φ-variation in several variables

Using classical techniques related to the so-called Hardy–Vitali variation, we present the class of X-valued functions of bounded \(\Phi \)-variation in several variables, where \((X,d,+ )\) is a metric semigroup. We exhibit some of the main properties of this class; among them, we show that this class can be made into a normed space and present a counterpart of the renowned Riesz’s Lemma for the case in which \(X=\mathbb {R}\) with its usual metric.