  We prove a multivariable approximate Carleman theorem on the determination of
complex measures on ${\mathbb{R}}^n$ and ${\mathbb{R}}^n_+$ by their moments.
This is achieved by means of a multivariable Denjoy--Carleman maximum principle
for quasi-analytic functions of several variables. As an application, we obtain
a discrete Phragm\'{e}n--Lindel\"{o}f-type theorem for analytic functions on
${\mathbb{C}}_+^n$.
