##
**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 |

### Keywords:

sequential convergence; integration with respect to polymeasures; Radon measure; Radon-Nikodým theorem; dominated convergence
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\textit{B. Jefferies}, Bull. Aust. Math. Soc. 32, 207--215 (1985; Zbl 0577.28002)

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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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