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Article

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

Journal of Mathematical Analysis and Applications. 2010. Vol. 369. No. 1. P. 82-93.
Chistyakov V., Tretyachenko Y.

Given a=(a1,…,an), b=(b1,…,bn)∈Rn with ab componentwise and a map f from the rectangle Iab=[a1,b1]×⋯×[an,bn] into a metric semigroup M=(M,d,+), denote by TV(f,Iab) the Hildebrandt–Leonov total variation of f on Iab, 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 fromIabinto M is such that the closure in M of the set{fj(x)}j∈Nis compact for eachx∈IabandC≡supj∈NTV(fj,Iab)is finite, then there exists a subsequence of{fj}j∈N, which converges pointwise onIabto a map f such thatTV(f,Iab)⩽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∗.