F. Riesz Theorem on harmonic and subharmonic functions

Question

In the book "Uniform Algebras and Jensen Measures" by T.W. Gamelin, p.39, says:

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By the F.Riesz Theorem, any subharmonic function $u$ in a neighborhood of a compact set $K$ in $C$ can be expressed in the form

$$u(z)= v(z) + \int log|z- \zeta|d\tau(\zeta),\quad z \in K$$

where $\tau$ is a positive measure supported on a compact neighborhood of $K$, and $v$ is harmonic in a neighborhood of $K$.

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Where can I find a reference for this result, especially for a proof? Can anyone provide a proof? Thank you.

Answer 1

A proof for the fundamental theorem on subharmonic functions may be found in Masatsugu Tsuji's "On F. Riesz' fundamental theorem on subharmonic functions" (Tohoku Math. J. (2) 4(2): 131-140 (1952)).