Uniqueness theorems for subharmonic functions in unbounded domains

A theorem of Carlson says that a holomorphic function of exponential growth in the half-plane cannot approach zero exponentially along the boundary unless it vanishes identically. This paper presents a generalization of this result for subharmonic functions in a Greenian domain $\Omega$, using the Martin boundary, minimal fine topology and PWB solution to the $h$-Dirichlet problem. Applications of the general theorem to specific choices of $\Omega$, such as the half-space and strip, are presented in later sections.