Q:

Equality of measures

I have two measures $\mu$ and $\nu$ supported on compacts in $\mbox{int } \mathbb{R}^{n}_+$. Are there some sufficiently general classes of such measures for which
$$
  \int\limits_{\mathbb{R}^n_+} \frac{\mu(dx)}{x_1^{z_1} x_2^{z_2}\cdots x_n^{z_n}} = \int\limits_{\mathbb{R}^n_+} \frac{\nu(dx)}{x_1^{z_1} x_2^{z_2}\cdots x_n^{z_n}}
$$
that holds for any $\Re z_1 > R$, $\Re z_2 > R$, ..., $\Re z_n > R$ and $R>1$ implies $\mu = \nu$?

A:

It's true for any regular complex Borel measures with compact support.  Let $K$ be the union of the supports of the two measures.  The linear span $V_0$ of the functions
$1/(x_1^{z_1} \ldots x_n^{z_n})$ for $\Re z_j > 0$ is dense in $C(K)$ by the Stone-Weierstrass Theorem.  Since your functions are of the form $f/(x_1 \ldots x_n)^R$ for $f \in V_0$ and
$(x_1 \ldots x_n)^R$ has no zeros on $K$, their linear span is also dense in $C(K)$.
So $\mu$ and $\nu$, corresponding to continuous linear functionals on $C(K)$, agree on a dense set and therefore are equal.

