  The classical Truncated Moment problem asks for necessary and sufficient
conditions so that a linear functional $L$ on $\mathcal{P}_{d}$, the vector
space of real $n$-variable polynomials of degree at most $d$, can be written as
integration with respect to a positive Borel measure $\mu$ on $\mathbb{R}^n$.
We work in a more general setting, where $L$ is a linear functional acting on a
finite dimensional vector space $V$ of Borel-measurable functions defined on a
$T_{1}$ topological space $S$. Using an iterative geometric construction, we
associate to $L$ a subset of $S$ called the \textit{core variety},
$\mathcal{CV}(L)$. Our main result is that $L$ has a representing measure $\mu$
if and only if $\mathcal{CV}(L)$ is nonempty. In this case, $L$ has a finitely
atomic representing measure, and the union of the supports of such measures is
precisely $\mathcal{CV}(L)$. We also use the core variety to describe the
facial decomposition of the cone of functionals in the dual space $V^{*}$
having representing measures. We prove a generalization of the Truncated
Riesz-Haviland Theorem of Curto-Fialkow, which permits us to solve a
generalized Truncated Moment Problem in terms of positive extensions of $L$.
These results are adapted to derive a Riesz-Haviland Theorem for a generalized
Full Moment Problem and to obtain a core variety theorem for the latter
problem.
