Definition of logarithmic capacity

Question

In the definition of logarithmic capacity of a compact set $E$ in the plane, the Robin constant is defined to be $V(E)=\inf\int_E\int_E \log\frac{1}{|z-w|} d\mu(z)d\mu(w)$ where $\inf$ is taken over all probability measures $\mu$ with supp$(\mu)\subset E$. Here is my question. Can infimum be taken not only over positive probability measures but over all "nice enough" signed measures with total variation one? say like $d\lambda= fdm$ where $f\in L^2(E)$ ,and $dm$ is Lebesgue measure. If so does that change our infimum meaning Robin constant ?if it changes, does it stay positive?