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 $K$.