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Given a map from a rectangle in the n-dimensional real Euclidean space into ametric semigroup, we introduce a concept of the total variation, whichgeneralizes a similar concept due to T. H. Hildebrandt (1963) for realfunctions of two variables and A. S. Leonov (1998) for real functions of nvariables, and study its properties. We show that the total variation has manyclassical properties of Jordan's variation such as the additivity, generalizedtriangle inequality and sequential lower semicontinuity. We prove two variantsof a pointwise selection principle of Helly-type, one of which is as follows: apointwise precompact sequence of metric semigroup valued maps on the rectangle,whose total variations are uniformly bounded, admits a pointwise convergentsubsequence.

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