Let
- $(\mathcal D(A),A)$ be a linear operator on $C_0(\mathbb R)$ (the space of continuous functions vanishing at infinity equipped with the supremum norm $\left\|\;\cdot\;\right\|_\infty$) such that $C_c^\infty(\mathbb R)$ is a core of $(\mathcal D(A),A)$
- $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$, $$\Gamma(f,g):=\frac12\left(A(fg)-fAg-gAf\right)\;\;\;\text{for }f,g\in C_c^\infty(\mathbb R)$$ and $$\mathcal E(f,g):=\int\Gamma(f,g)\:{\rm d}\mu\;\;\;\text{for }f,g\in C_c^\infty(\mathbb R)$$
Assuming $$\int Af\:{\rm d}\mu=0\;\;\;\text{for all }f\in C_c^\infty(\mathbb R)\tag1$$ and $$\int fAg\:{\rm d}\mu=\int gAf\:{\rm d}\mu\;\;\;\text{for all }f,g\in C_c^\infty(\mathbb R),\tag2$$ it's easy to see that $$\mathcal E(f,g)=-\int fAg\:{\rm d}\mu=-\int gAf\:{\rm d}\mu\;\;\;\text{for all }f,g\in C_c^\infty(\mathbb R)\tag3.$$
Are we able to extend the definition of $\mathcal E$, preserving $(3)$, for $f\in\mathcal D(A)$ and $g\in C_c^\infty(\mathbb R)$?
Since $f\in\mathcal D(A)$ and $C_c^\infty(\mathbb R)$ is a core of $(\mathcal D(A),A)$, we should find a $(f_n)_{n\in\mathbb N}\subseteq C_c^\infty(\mathbb R)$ with $$\left\|f_n-f\right\|_\infty+\left\|Af_n-Af\right\|_\infty\xrightarrow{n\to\infty}0.\tag4$$ By submultiplicativity of $\left\|\;\cdot\;\right\|_\infty$, we should have $$\left|\mathcal E(f_n,g)-\mathcal E(f,g)\right|\le\frac12\left(\left\|A(f_ng)-A(fg)\right\|_\infty+\left\|f_n-f\right\|_\infty\left\|Ag\right\|_\infty+\left\|g\right\|_\infty\left\|Af_n-Af\right\|_\infty\right)\tag5$$ and by $(4)$ the last two terms on the right-hand side tend to $0$ as $n\to\infty$.
Do we have $\left\|A(f_ng)-A(fg)\right\|_\infty\xrightarrow{n\to\infty}0$ too?
On the other hand, $\mathcal E(f_n,g)=-\mu(f_nAg)=-\mu(gAf_n)$ by $(3)$ for all $n\in\mathbb N$ and we should easily obtain $$\left|\mu(f_nAg)-\mu(fAg)\right|+\left|\mu(gAf_n)-\mu(gAf)\right|\xrightarrow{n\to\infty}0$$ by $(4)$ and the dominated convergence theorem. Thus, we would be able to conclude the desired claim.
It might be useful to observe that $f_ng,fg\in C_c^\infty(\mathbb R)$ (since $g\in C_c^\infty(\mathbb R)$) for all $n\in\mathbb N$.