Transformations of Moment Functionals

TL;DR

This paper explores how measure transformations affect moment functionals, providing characterizations and representations of moments functionals on compact, path-connected sets, linking them to Hausdorff moment problems and measurable functions.

Contribution

It offers new characterizations of moment functionals via measure transformations and establishes their representations through continuous and measurable functions.

Findings

Existence of a measurable function g for characterizing K-moment functionals.

Representation of moment functionals via continuous extensions and Hausdorff moment functionals.

Every moment functional can be represented by a composition with a measurable function and Lebesgue measure.

Abstract

In measure theory several results are known how measure spaces are transformed into each other. But since moment functionals are represented by a measure we investigate in this study the effects and implications of these measure transformations to moment funcationals. We gain characterizations of moments functionals. Among other things we show that for a compact and path connected set $K\subset\mathbb{R}^n$ there exists a measurable function $g:K\to [0,1]$ such that any linear functional $L:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}$ is a $K$-moment functional if and only if it has a continuous extension to some $\overline{L}:\mathbb{R}[x_1,\dots,x_n]+\mathbb{R}[g]\to\mathbb{R}$ such that $\tilde{L}:\mathbb{R}[t]\to\mathbb{R}$ defined by $\tilde{L}(t^d) := \overline{L}(g^d)$ for all $d\in\mathbb{N}_0$ is a $[0,1]$-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function $f:[0,1]\to K$ independent on $L$ such that the representing measure $\tilde{\mu}$ of $\tilde{L}$ provides the representing measure $\tilde{\mu}\circ f^{-1}$ of $L$. We also show that every moment functional $L:\mathcal{V}\to\mathbb{R}$ is represented by $\lambda\circ f^{-1}$ for some measurable function $f:[0,1]\to\mathbb{R}^n$ where $\lambda$ is the Lebesgue on $[0,1]$.