多元有限全变差映射与Helly型选择原理
Maps of several variables of finite total variation and Helly-type selection principles
摘要：
给定n维实欧几里得空间中的一个矩形到度量半群的映射，引入了全变差的概念，推广了T. H. Hildebrandt（1963）对二元实函数和A. S. Leonov（1998）对n元实函数的性质进行了研究.证明了全变差具有Jordan变差的许多经典性质，如可加性、广义三角不等式和序列下半连续性。本文证明了Helly型点态选择原理的两个变形，其中之一是：矩形上度量半群值映射的点态预紧序列，其全变差一致有界，存在点态收敛子序列.
英文摘要：
Given a map from a rectangle in the n-dimensional real Euclidean space into a metric semigroup, we introduce a concept of the total variation, which generalizes a similar concept due to T. H. Hildebrandt (1963) for real functions of two variables and A. S. Leonov (1998) for real functions of n variables, and study its properties. We show that the total variation has many classical properties of Jordan's variation such as the additivity, generalized triangle inequality and sequential lower semicontinuity. We prove two variants of a pointwise selection principle of Helly-type, one of which is as follows: a pointwise precompact sequence of metric semigroup valued maps on the rectangle, whose total variations are uniformly bounded, admits a pointwise convergent subsequence.
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