Let $M$ be a compact Riemann surface. Let $\Lambda$ be a differentiable real $2$-form of integral one. Let $G$ be the Green function associated to $\Lambda$, i.e. $G: M \times M \to \mathbb R \cup \{\infty \}$ is symmetric, $G_x(y) = G(x,y)$ is integrable for every $x \in M$ (with respect to any positive differentiable $2$-form), $\int G_x(y) \Lambda(dy) = 0$ for every $x \in M$ and $$\Delta G_x = -\delta_x + \Lambda .$$
It is known that $G$ is bounded from below. Denote by $\mathcal P(M)$ the space of probabilities in $M$. We can define the functional $$I : \mathcal P(M) \to \mathbb R \cup \{ \infty \}$$ $$\mu \mapsto \int_{M \times M} G(x,y) \, \mu(dx) \, \mu(dy).$$
I would like to prove that $I$ is convex. One possible approach is to try to prove that $$\frac{1}{2} \int \mu(dx) G(x,y) \mu(dy) + \frac{1}{2} \int \nu(dx) G(x,y) \nu(dy) \geq \int \mu(dx) G(x,y) \nu(dy)$$ for every $\mu, \nu \in \mathcal P(M)$. We can see that this is true for differentiable $2$-forms. To complete the proof I would need the following to be true: For any $\mu \in \mathcal P(M)$, there exists a sequence $\{\mu_n\}$ of differentiable $2$-forms such that $$\mu_n \to \mu$$ weakly and $$I(\mu_n) \to I(\mu).$$
I would really appreciate if someone could tell me how to prove the existence of such sequence or, in any case, another way to prove the convexity of $I$.
