Suppose $D$ is a bounded domain of $\mathbb{R}^{n}$ with $n>1$ and $E$ a subset of its boundary. We know that if $E$ has capacity zero I.e. it is a polar set , then the harmonic measure of $E$ with respect to $D$ is also zero: $\omega(x,D)(E)=0$ for all $x\in D$. Now suppose the capacity of $E$ is $<\epsilon$. Can we say that $\omega(x,D)(E)<\epsilon?$ ($\epsilon$ is a positive arbitrary number).



Capacity and harmonic measure
We can say that for the Martin capacity, see <https://projecteuclid.org/euclid.aop/1176988187>


For the classical capacity there is a correction factor of the minimum 
of the potential kernel from $x$ to $D$ as explained in that reference. 


All this is in dimension 3 and higher. There are some special effects in 2D. 


