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Radon polymeasures. (English) Zbl 0577.28002

Polymeasures are set functions defined on an algebra of subsets of a product space such that the set function is additive, but only componentwise \(\sigma\)-additive. Examples arise in the treatment of certain Feynman integrals. The set function m defined on the family of all products of Borel sets in \({\mathbb{R}}\) by \[ m(A\times B)=\int_{A}\int_{B}\phi (x)\exp [i(x-y)^ 2]\psi (y) dx dy \] where \(\phi\) and \(\psi\) belong to \(L^ 2({\mathbb{R}})\) is a bimeasure. It extends uniquely to an additive set function on the algebra generated by all such product sets in \({\mathbb{R}}\times {\mathbb{R}}.\) However, if either \(\phi\) or \(\psi\) does not belong to \(L^ 1({\mathbb{R}})\), then m is unbounded on this algebra, so it cannot be the restriction of a measure on the generated \(\sigma\)-algebra. - The purpose of this note is to show that there is a useful theory of integration with respect to polymeasures of a certain class. These include those with finitely many components such that the associated (possibly infinite) variation set function is the restriction of a Radon measure on the product space. It is shown that with such an approach, a type of Radon-Nikodým theorem and dominated convergence result are true.

MSC:

28A35 Measures and integrals in product spaces
28A15 Abstract differentiation theory, differentiation of set functions
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28A25 Integration with respect to measures and other set functions
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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References:

[1] DOI: 10.1007/BF02417887 · Zbl 0399.46032
[2] Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures (1973) · Zbl 0298.28001
[3] Fremlin, Topological Riesz Spaces and Measure Theory (1974)
[4] DOI: 10.2307/2044201 · Zbl 0456.28003
[5] DOI: 10.1090/S0002-9904-1947-08755-3 · Zbl 0032.35702
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