The mean value for infinite volume measures, infinite products and heuristic infinite dimensional Lebesgue measures

TL;DR

This paper develops a unified framework for defining and analyzing mean values on infinite product spaces with infinite measures, extending classical concepts to infinite-dimensional settings like Hilbert spaces.

Contribution

It introduces a systematic approach to limits of means on infinite measured spaces and defines a heuristic Lebesgue measure in infinite dimensions, preserving key invariance properties.

Findings

Limits of means extend to infinite measures and spaces.

A translation and scaling invariant mean is constructed in infinite dimensions.

The approach generalizes classical finite-dimensional measure concepts.

Abstract

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some known examples coming from the theory of metric measured spaces and also from oscillatory integrals are obtained as limits of means with respect to finite measures. Then, we explore in a systematic way the limit of means of the type $$ \lim \frac{1}{\mu(U_n)} \int_{U_n} f d\mu$$ where $\mu$ is a a $\sigma-$finite Radon measure $\mu.$ In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (but principally on a Hilbert space) is defined. This last object is shown to be invariant by translation, scaling and restriction.