  We consider the problem of finding a (non-negative) measure $\mu$ on
$\mathfrak{B}(\mathbb{C}^n)$ such that $\int_{\mathbb{C}^n}
\mathbf{z}^{\mathbf{k}} d\mu(\mathbf{z}) = s_{\mathbf{k}}$, $\forall
\mathbf{k}\in\mathcal{K}$. Here $\mathcal{K}$ is an arbitrary finite subset of
$\mathbb{Z}^n_+$, which contains $(0,...,0)$, and $s_{\mathbf{k}}$ are
prescribed complex numbers (we use the usual notations for multi-indices).
There are two possible interpretations of this problem. At first, one may
consider this problem as an extension of the truncated multidimensional moment
problem on $\mathbb{R}^n$, where the support of the measure $\mu$ is allowed to
lie in $\mathbb{C}^n$. Secondly, the moment problem is a particular case of the
truncated moment problem in $\mathbb{C}^n$, with special truncations. We give
simple conditions for the solvability of the above moment problem. As a
corollary, we have an integral representation with a non-negative measure for
linear functionals on some linear subspaces of polynomials.
