[Banach Space Bulletin Board] Abstract of a paper by Vyacheslav V. Chistyakov, Svetlana A. Chistyakova
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
participants (1)
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Bentuo Zheng (bzheng)