On evaluating the measure of strong projections in infinite dimension

TL;DR

This paper explores the measure of projections of finite-dimensional sets in infinite-dimensional spaces, establishing conditions under which measure continuity holds and introducing the concept of strong projections to extend these results.

Contribution

It introduces the concept of strong projections in infinite-dimensional measure spaces and demonstrates measure continuity for these projections with Lebesgue measure.

Findings

Measure continuity holds for discrete measures.

Lebesgue measure is not continuous with standard projections.

Continuity is restored using strong projections.

Abstract

Projections of finite dimensional sets and their measures are investigated in infinite-dimensional power measure spaces. The starting point is the known algebraic formula, expressing \ the $y$-projection of a finite-dimensional set $a$ as a Boolean supremum of certain finite geometrical transformations of $a$ in the infinite-dimensional power space. This Boolean supremum somewhat unusual in classical measure theory because, it is different, in general, from the usual union of sets. The paper investigates the problem whether the power measure in the infinite-dimensional measure space is continuous with respect to the forementioned Boolean supremum. If so, then this continuity leads to a simple formula for calculating the measure of the projection of $a.$ It is shown that the answer concerning this continuity is affirmative for discrete measures but false for the Lebesgue measure, for example. However, it is proved that if the concept of the $y$-projection of $a$ is replaced by that of the so-called \textit{strong} $y$-\textit{projection} of $a,$ then the Lebesgue measure becomes continuous in this context and the value of the corresponding real supremum is exactly the measure of the foregoing strong $y $-projection. In this paper, the tools of the classical analysis are adapted to handle measures on Boolean algebras.