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Pointwise selection theorems for metric space valued bivariate functions

Journal of Mathematical Analysis and Applications. 2017. Vol. 452. No. 2. P. 970-989.
Vyacheslav V. Chistyakov, Svetlana A. Chistyakova.

We introduce a pseudometric TV on the set M^X of all functions mapping a rectangle X on the plane R^2 into a metric space M, called the total joint variation. We prove that if two sequences {fj} and {gj} of functions from M^X are such that {fj} is pointwise precompact on X, {gj} is pointwise convergent on X with the limit g∈M^X, and the limit superior of TV(fj, gj) as j→∞ is finite, then a subsequence of {fj} converges pointwise on X to a function f∈M^X such that TV(f, g ) is finite. One more pointwise selection theorem is given in terms of total ε-variations (ε >0), which are approximations of the total variation as ε →0.