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The joint modulus of variation of metric space valued functions and pointwise selection principles

arxiv.org. math. Cornell University, 2016. No. 1601.07298.
Given a subset T of real numbers and a metric space M, we introduce a nondecreasing sequence {v_n} of pseudometrics on the set M^T of all functions from T into M, called the joint modulus of variation. We prove that if two sequences of functions {f_j} and {g_j} from M^T are such that {f_j} is pointwise precompact, {g_j} is pointwise convergent, and the limit superior of v_n(f_j,g_j) as j → ∞ is o(n) as n → ∞, then {f_j} admits a pointwise convergent subsequence whose limit is a conditionally regulated function. We illustrate the sharpness of this result by examples (in particular, the assumption on the limsup is necessary for uniformly convergent sequences {f_j} and {g_j}, and ‘almost necessary’ when they converge pointwise) and show that most of the known Helly-type pointwise selection theorems are its particular cases.