Let $D$ be a bounded domain of $\mathbb{R}^{m}$ with $m>1$, and $u$ a subharmonic function on $D$. Let $u_{\epsilon} $ be a sequence of smooth subharmonic functions on $D_{\epsilon}$ (the set of elements of $D$ having a distance bigger than $\epsilon$ from the complement of $D$) that decreases to $u$ pointwise. Let $\mu$ be the Riesz measure associated to $u$ and $\mu_{\epsilon}$ the Riesz measure associated to $u_{\epsilon} $. Let $E\subset D$ be a Borel set.
My question is: suppose $E$ is a $\mu_{\epsilon}-$null set for all $\epsilon>0$, can we conclude that $E$ is $\mu-$null set?
No. For example $m=2$, $D$ the unit disk, $u(x,y)=|y|$, $E=\{(t,0):|t|\le1/2\}$.
(That's really just an example for $m=1$ with an extra variable added since you specified $m>1$; for $m=1$ take $u(t)=|t|$, $D=(-1,1)$, $E=\{0\}$.)