I have a question about theory of Markov processes.
Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $E$ be a Hausdorff topological space and $\mathcal{B}(E)$ be its Borel $\sigma$-algebra and $m$ denotes a $\sigma$-finite positive measure on $(E,\mathcal{B}(E))$. Let $X$ be a Markov process in a continuous time with value in the set $E$.
We assume the following conditions holds:
- $\mathcal{B}(E)=\sigma(C(E))$ where $C(E)$ denotes the set of all continuous ($\mathbb{R}$-valued) function on $E$
- For all $\omega \in \Omega$, $t \mapsto X_{t}(\omega)$ is right continuous on $[0,\infty[$
- $P_{x}[X_{0}=x]=1\quad({}^{\forall} x\in E)$
In the above-mentioned setting, we define \begin{align*} R_{\alpha}f(x):=E_{x}\left[ \int_{0}^{\infty} e^{-\alpha t} f(X_{t})dt\right]\quad(\alpha>0,x \in E, f \in \mathcal{B}(E)^{+}) \end{align*}
My question:
Since $m$ is $\sigma$-finite, there exists $\varphi \in \mathcal{B}(E)$ such that $\varphi(x)>0$ for all $x \in E$ . For this $\varphi$, $R_{1}\varphi(x)>0,\,x \in E$ is hold?
My attempt:
Case1: $\varphi$ is continuous
Since $t \mapsto X_{t}(\omega)$ is right continuous and $P_{x}[X_{0}=x]=1\quad({}^{\forall} x\in E)$, it is obvious that $R_{1}\varphi(x)>0$ forall $x \in E$ .
Case2 $\varphi$ is not continuous
Although I think $\varphi$ is a kind of continuous function by the condition $\mathcal{B}(E)=\sigma(C(E))$, I don't know how to the proper way to use this condition.
Thank you in advance.
Yes, $R_1 \varphi>0$ holds true.
Suppose that $R_1 \varphi(x)=0$ for some $x \in E$. Then
$$\mathbb{E}^x \left( \int_0^{\infty} e^{- t} \varphi(X_t) \, dt \right)=0$$
implies
$$e^{- t} \varphi (X_t)=0$$
almost everywhere (since the mapping $(\omega,t) \mapsto e^{- t} \varphi(X_t(\omega))$ is non-negative). Hence,
$$\varphi(X_t)=0$$
almost everywhere. This is a contradiction to our assumption that $\varphi>0$.