On Functions Which are Superharmonic for a Markov Process

Let $X=(x_t,\zeta,\mathcal{M}_t,{\mathbf P}_x)$ be a standard Markov process on a locally compact separable Hausdorff space $(E,\mathcal{O})$. An almost Borel measurable function $f(x):E \to({-\infty,+\infty}]$ is called superharmonic if it satisfies the following conditions: a) it is intrinsically continuous; b) ${\mathbf M}_x f(x(\tau_G))\leqq f(x)$ for any $x\in E$ and any open set $G$ with a compact closure, where $\tau_G$ is the hitting time for the set $E\setminus G$.
The main results are stated in Theorems 1 and 2. In these theorems $S$ denotes the set of $x\in E$ for which $x_t$ coincides with $x$ (${\mathbf P}_x$ almost surely) during a positive random time interval $[0,\delta]$; the symbol $\mathcal{U}$ denotes any open base of $\mathcal{O}$, and $\mathcal{V}$ is the class of all sets $U$ of the type $U\in\mathcal{U}$ or $U=V\setminus S$, where $V\in\mathcal{U}$.
Theorem 1. {\it A non-negative almost Borel function$f(x)$, $x\in E$, is superharmonic if and only if it is intrinsically continuous and
$$ M_x f\left({x\left({\tau_U}\right)}\right)\leqq f(x) $$
for any$x\in E$ and any $U\in\mathcal{V}$.}
Theorem 2. {\it A non-negative function $f(x)$, $x\in E$, which is semicontinuous from below is superharmonic if and only if it satisfies the condition $(*)$ for any $x\in E$ and any $U\in\mathcal{U}$.}