This is an announcement for the paper “The joint modulus of variation of metric space valued functions and pointwise selection principles” by Vyacheslav V. Chistyakov, Svetlana A. Chistyakova.
Abstract: Given $T\subset\mathbb{R}$ and a metric space $M$, we introduce a nondecreasing sequence of pseudo metrics ${\mu_n}$ on $MT$ (the set 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 $MT$ are such that $(f_j)$ is pointwise precompact, $(g_j)$ is pointwise convergent, and the limit superior of $\mu_n(f_j, g_j)$ as $j\to\infty$ is $o(n)$) as $n\to\infty$, 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 lim sup 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. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1601.07298