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

Given a = ( a 1 , … , a n ) , b = ( b 1 , … , b n ) ∈ R n with a < b componentwise and a map f from the rectangle I a b = [ a 1 , b 1 ] × ⋯ × [ a n , b n ] into a metric semigroup M = ( M , d , + ) , denote by TV ( f , I a b ) the Hildebrandt–Leonov total variation of f on I a b , 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 { f j } j ∈ N of maps from I a b into M is such that the closure in M of the set { f j ( x ) } j ∈ N is compact for each x ∈ I a b and C ≡ sup j ∈ N TV ( f j , I a b ) is finite, then there exists a subsequence of { f j } j ∈ N , which converges pointwise on I a b to a map f such that TV ( f , I a b ) ⩽ C . 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 ∗ .