This is probably easy questions but I could not find a good explanation for myself.
When I'm reading about polynomial complex dynamics, I found a technical term measure-valued Laplacian for a Green's function associated to a polynomial.
Many of books/authors say that it's Laplacian $\color{blue}{in\,\, the\,\, sense \,\,of\,\, distribution}$.
My question is what does it means to be Laplacian in the sense of distribution?
Plus, this defines a measure so-called $\color{blue}{harmonic\,\,measure}$. Actually, I have seen these terms used interchangeably with $\color{blue}{equilibrium\,\,measure}$.
Can someone explain me these notion of measures? I think I understand that the equilibrium measure means in the sense of potential theory which minimizes the energy integral. But, the harmonic measure I'm not quite get it (both in terms of definition and usage).
Distributions are roughly speaking generalized functions. Measures are particular examples of distributions. This is a technical subject, and I won't try to give here a precise definition of distributions.
One of the greatest advantage of distributions is that you can always differentiate a distribution and get another distribution. Here, Laplacian in the sense of distribution means: take a function $u$, think of it as a distribution, apply the the Laplacian, and you get a distribution $\Delta u$. It turns out that if $u$ is a subharmonic function, this procedure gives you in fact a measure $\Delta u$.
The equilibrium measure, as you say, is defined as maximizing the energy. The harmonic measure of a compact $K \subset \mathbb C$ is defined as $\Delta u$, where $u$ is the Green function associated to $K$. It is subharmonic, and so $\Delta u$ is indeed a measure. Again, it turns out that these two things are the same.
I don't know what you've been reading to prompt these questions, by Ransford's book "Potential theory in the complex plane" is a very good place to find more detailed answers.