Projectively bounded Fréchet measures

Blei, Ron

doi:10.1090/S0002-9947-96-01625-X

Projectively bounded Fréchet measures
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by Ron C. Blei PDF

Trans. Amer. Math. Soc. 348 (1996), 4409-4432 Request permission

Abstract:

A scalar valued set function on a Cartesian product of $\sigma$-algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequality. It is shown that feasibility of products of Fréchet measures is linked to the projective boundedness property. All Fréchet measures in a two dimensional framework are projectively bounded, while there exist Fréchet measures in dimensions greater than two that are projectively unbounded. A basic problem is considered: when is a Fréchet measure projectively bounded? Some characterizations are stated. Applications to harmonic and stochastic analysis are given.

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