Transformations of Moment Functionals

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 $\tildeμ$ of $\tilde{L}$ provides the representing measure $\tildeμ\circ f^{-1}$ of $L$. We also show that every moment functional $L:\mathcal{V}\to\mathbb{R}$ is represented by $λ\circ f^{-1}$ for some measurable function $f:[0,1]\to\mathbb{R}^n$ where $λ$ is the Lebesgue on $[0,1]$.