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 $x\in E$?



A simple clarification on Riesz decomposition theorem
(You need a minus sign in front of $\int\_E K(x-\zeta)\,d\mu(\zeta)$.)


The integral $\,-\!\!\int\_E K(x-\zeta)\,d\mu(\zeta)$ is subharmonic throughout $D$ and harmonic in $D\setminus E$. If $u$ admits a harmonic majorant in $D$, hence a least such majorant (call it $k$), then $h$ (which depends on $E$) can be expressed as $\,-\!\!\int\_{E^c} K(x-\zeta)\,d\mu(\zeta)+k(x)$ for $x\in D$.


