  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
$[0,1]$.
