So I'm seeing a Dirichlet form written as
$\mathscr{E}(f,f) = \frac{1}{2} \sum_{x,y} |f(x)-f(y)|^2 K(x,y)\pi(x)$
where $K(x,y)$ is the probability of taking a step to state $y$ from $x$.
And the conventional way of writing it seems to be
$\mathscr{E}(f) = \int \left|\triangledown f\right|^2 d\mu. $
How come the two expressions are equivalent. I just don't see how $K(x,y)\pi(x)$ could be analogous to $d\mu$ like $(f(x)-f(y))^2$ is analogous to $|\triangledown f|^2.$ If it were $\pi$, I could understand, but why are they using $K(x,y)\pi(x)?$
I hope this isn't a stupid question (I feel like it is).