Let $(X,m)$ be a locally compact measure space countable at infinity. Suppose we have a bilinear form $a:H \times H \to \mathbb{R}$ on a Hilbert space $H \subset L^2(X)$.
The form is coercive and bounded, and satisfies the property: if $u \in H$, then $u^+, u^- \in H$ and $a(u^+,u^-) =0$. That is, it preserves positivity. Furthermore, we have $C_0(X) \cap H \subset C_0(X)$ is dense wrt. the sup norm, but $C_0(X) \cap H \subset H$ is not dense. Hence we are not in the situation of a regular form. And $a$ may not be a Dirichlet form.
The notation $C_0(X)$ means compactly supported continuous functions.
Is it possible to still define a notion of capacity and quasicontinuous functions?
