Let $D$ be a domain of $\mathbb{R}^{m}$ and let $K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions", vol. 1, pg 104) if $u$ is subharmonic on $D$, then there is a unique Borel measure $\mu$ such that for all compact $E$ in $D$ we have $$u(x)=\int_{E}K(x-\zeta)d\mu(\zeta)+h(x)$$ where $h$ is harmonic on the interior of $E$.

Here is my question: is this equation valid for all $x\in D$ or all $x\in E$?

$u$ can be represented in the above form in $D$ with $h$ that is harmonic in the interior of $E$ (and subharmonic in the rest of the domain).The key point is that for every compact subset of $D$, the Riesz measure is finite, so the potential is well-defined but this may be not so in the entire domain $D$. $\endgroup$ – fedja Sep 22 '20 at 1:57