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On integration in Banach spaces. VIII: Polymeasures. (English) Zbl 0688.28002

A vector d-polymeasure (d a positive integer) is a separately countably additive set function defined on the Cartesian product of d rings with value in a vector space. Uniform vector d-polymeasures are defined naturally. An interesting case is when the rings are \(\delta\)-rings and the values are operators from the projective tensor product of d Banach spaces into another Banach space; in this case the operator-valued, countably additive in strong operator topology, polymeasures are considered. The notion is non-trivial, because there exist such objets which are not restrictions of the usual operator-valued measures from the \(\delta\)-ring generated by the Cartesian product of the given \(\delta\)- rings. The author studies the properties of polymeasures and their semivariations in general, and for the case of uniformness, with the aim of extending his theory of integration [see Parts I-VII in this case: same journal 20(95), 511-536 (1970; Zbl 0215.201), 20(95), 680-695 (1970; Zbl 0224.46050); 29(104), 478-499 (1979; Zbl 0429.28011); 30(105), 259-279 (1980; Zbl 0425.28006); 30(105), 610-628 (1980; Zbl 0506.28004); 35(110), 173-187 (1985; Zbl 0628.28007); 38(113), 433-449 (1988; Zbl 0674.28003), respectively]. The key problem of existence of control polymeasures is not solved, and this affects further developments.
Reviewer: Gr.Arsene

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
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