Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

Given a=(a₁,…,an), b=(b₁,…,bn)∈Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), denote by the Hildebrandt-Leonov total variation of f on , which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}_j∈Nof maps frominto M is such that the closure in M of the set{fj(x)}_j∈Nis compact for eachandis finite, then there exists a subsequence of{fj}_j∈N, which converges pointwise onto a map f such that. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M^∗.