Abstract
Let
μ
be a given Borel measure on $\K\subseteq\R^n$ and let
y=(
y
α
)
, $\alpha\in\N^n$, be a given sequence. We provide several conditions linking
y
and the moment sequence
z=(
z
α
)
of
μ
, for
y
to be the moment sequence of a Borel measure
ν
on $\K$ which is absolutely continuous with respect to
μ
and such that its density is in $L_\infty(\K,\mu)$. The conditions are necessary and sufficient if $\K$ is a compact basic semi-algebraic set, and sufficient if $\K\equiv\R^n$. Moreover, arbitrary finitely many of these conditions can be checked by solving either a semidefinite program or a linear program with a single variable