Geodesic
Uniform boundedness of a~family of set functions
Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 855-861

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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. 
@article{MZM_1978_23_6_a7,
     author = {M. Kh. Khafizov},
     title = {Uniform boundedness of a~family of set functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {855--861},
     publisher = {mathdoc},
     volume = {23},
     number = {6},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_6_a7/}
}
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M. Kh. Khafizov. Uniform boundedness of a~family of set functions. Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 855-861. http://geodesic.mathdoc.fr/item/MZM_1978_23_6_a7/