the truncated moment problem seeks conditions on an n-dimensional multisequence of degree $m$, $y \equiv (y_i)_{|i| ≤ m}$, such that there exists a positive borel measure $\mu$ on $\mathbb{r}^n$ satisfying $y_i = \int \xi d \mu \, (|i| ≤ m)$ (where $x = (x_1, \ldots, x_n)$, $i = (i_1, \ldots, i_n)$). in previous work we associated to $y$ an algebraic variety in $\mathbb{r}^n$ , the core variety $v = v(y)$, and showed that if $v$ is nonempty, then the riesz functional $l$ corresponding to $y$ is strictly v-positive, i.e., if $p(x) := \sigma a_i x_i \, (|i| ≤ m)$ is nonnegative on $v$, and $p|_v$ is not identically $0$, then $l(p) := \sigma a_i y_i > 0$. in current work with g. blekherman, we prove that if $l$ is strictly $k$-positive for any closed subset $k$ of $\mathbb{r}^n$, then $y$ has a representing measure $\mu$ (as above) whose support is contained in $k$. as a consequence, we prove that $y$ has a representing measure if and only if $v(y)$ is nonempty, in which case $v(y)$ coincides with the union of the supports of all representing measures. as a corollary, we obtain a new proof of the bayer-teichmann theorem on multivariable cubature.