Lebesgue-Feynman Measures on Infinite Dimensional Spaces

Abstract

The Lebesgue-Feynman measure on a linear space E is a generalized measure on E which is defined as a linear functional μ on some linear space F(E,ℂ) of functions E → ℂ which is invariant with respect to the shift on any vector. There exists two different approach to definition of Lebesgue-Feynman measure. The first definition is given by Feynman formula and described in the works (Smolyanov and Shavgulidze 2015; Sadovnichiy et al., Doklady Math. 86(2), 644–647, 2012, Doklady Math. 93(1), 46–48, 2016). In this paper we consider the second approach which is based on the definition of translation-invariant function of a set on some ring of elementary subsets. As it is well-known there exist Sobolev-Schwartz distributions which do not correspond to any function of a set. In contrast, the Lebesgue-Feynman generalized measure as we show below is closely connected to the shift-invariant function of a set which is determined on the ring of elementary subsets.