# Existence of Infinite Product Measures

## TL;DR

This paper presents an elementary construction of infinite product measures for any sequence of measure spaces using outer measure techniques, simplifying the theory and its computational aspects.

## Contribution

It introduces a new elementary approach to constructing infinite product measures without restrictions, and simplifies the associated $L_{p}$ spaces for easier analysis.

## Key findings

- Constructed product measures for arbitrary measure spaces.
- Simplified the $L_{p}$ spaces of these measures.
- Provided a computational framework for infinite dimensional integration.

## Abstract

A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the existence of product measures in an elementary manner. Moreover, the $L_{p}$ spaces of this measures are simplified in terms of finite product measures following the approach of [21]. This decomposition simplifies infinite dimensional integration and gives to this theory a computational framework.