We employ positivity of Riesz functionals to establish representing measures
(or approximate representing measures) for truncated multivariate moment
sequences. For a truncated moment sequence $y$, we show that $y$ lies in the
closure of truncated moment sequences admitting representing measures supported
in a prescribed closed set $K \subseteq \re^n$ if and only if the associated
Riesz functional $L_y$ is $K$-positive. For a determining set $K$, we prove
that if $L_y$ is strictly $K$-positive, then $y$ admits a representing measure
supported in $K$. As a consequence, we are able to solve the truncated
$K$-moment problem of degree $k$ in the cases: (i) $(n,k)=(2,4)$ and $K=\re^2$;
(ii) $n\geq 1$, $k=2$, and $K$ is defined by one quadratic equality or
inequality. In particular, these results solve the truncated moment problem in
the remaining open cases of Hilbert's theorem on sums of squares.