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 $E$.

I have two questions:

1) Suppose $E$ is a compact of $D$ with no interior. Then what happens to the above formula? Does it still hold?

2) Is it true that the function $h$ is given by the integral of $u$ over the boundary of $\partial E$ of $E$ with respect to the harmonic measure?