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 $[0,1]$.
Categories: math.FA