A uniqueness theorem for subharmonic functions of finite order

Let $u$ and $v$ be subharmonic functions of finite order on $\mathbf R^m$. The main theorem of this paper shows that, if $u\leqslant v$, the relation "$\leqslant$" is preserved, in a certain sense, for mass distributions $\mu_u$ and $\mu_v$. This result yields new uniqueness theorems for both subharmonic and entire functions on the complex plane. Corollaries include a broad class of sufficient conditions for the completeness of systems $\{e^{\lambda_nz}\}$ of exponential functions in a complex domain $G$. The conditions for completeness are stated entirely in terms of the distribution of the points of the sequence $\{\lambda_n\}$ in the neighborhood of infinity and in terms of the geometric properties (mixed areas) of $G$.