Local property of the generator of a Feller-Dynkin I have some trouble proving the following :
Let $L:D_L \to C_0(\mathbb{R}^+)$ the generator of a Feller-Dynkin semigroup $(P_t)_t$ of a strong continuous Markov process $(X_t)_t$ such as for all probability $\mu$ on $\mathbb{R}^+$, $\mathbb{P}_\mu[X_t \in \mathbb{R}^+]=1$. We also have $C^2_0(\mathbb{R}^+) \subset D_L$.
I'd like to prove this : for $f,g \in D_L$, if $f=g$ on a neighbourhood of $x\in \mathbb{R}^+ $ then $Lf(x)=Lg(x)$.
Some help would be appreciated.
 A: Showing that for $f,g \in D_L$ if $f = g$ on a neighbourhood of $x \in \mathbb{R}^+$ then $Lf(x) = Lg(x)$ is equivalent to showing that for $h \in D_L$ if $h \equiv 0$ in a neighbourhood of $x \in \mathbb{R}^+$ then $Lh(x) = 0$.
Denote by $V$ the open neighbourhood of $x \in \mathbb{R}^+$ on which $h \equiv 0$. By density, we can suppose that $h$ has compact support.
Let $\varepsilon > 0 $ such that $B\left(x, \varepsilon\right) := \left\{y \in \mathbb{R}^+ : \left|y - x\right| < \varepsilon\right\} \subset V$. By continuity of $\left(X_t\right)_{t\geq 0}$, for almost all $\omega \in \Omega$ there exists $\delta\left(\omega\right) > 0$ such that $\left|X_t\left(\omega\right) - x\right| < \varepsilon$ for all $t \leq \delta\left(\omega\right)$. This implies that $h\left(X_t\left(\omega\right)\right) = 0$ for all $t \leq \delta\left(\omega\right)$ and that $t^{-1} h\left(X_t\right) \to 0$ as $t \downarrow 0$ almost surely. Then by dominated convergence,
$$Lh(x) = \lim_{t\downarrow 0}t^{-1}\left(P_th(x) - h(x)\right) =  \lim_{t\downarrow 0} \mathbb{E}_x\left[t^{-1}h\left(X_t\right)\right] = 0.$$
