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This article is cited in 1 scientific paper (total in 1 paper)
Uniform boundedness of a family of set functions
M. Kh. Khafizov Elabuga Pedagogical Institute
Abstract:
Let $\Sigma$ be a ring of sets, $X$ a normed space, $\mu_\alpha:\Sigma\to X$ ($\alpha\in\Lambda$) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family $\{\mu_\alpha\}$$\{\mu_\alpha\}$ is uniformly bounded on $\Sigma$ if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a~part of some set in~$\Sigma$. An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a~family of triangular functions is preserved in passing from a~ring to the $\sigma$-ring it generates.
Received: 29.06.1976
Citation:
M. Kh. Khafizov, “Uniform boundedness of a family of set functions”, Mat. Zametki, 23:6 (1978), 855–861; Math. Notes, 23:6 (1978), 469–473
Linking options:
https://www.mathnet.ru/eng/mzm8186 https://www.mathnet.ru/eng/mzm/v23/i6/p855
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Abstract page: | 245 | Full-text PDF : | 239 | First page: | 2 |
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