Let $X=(X_t)_{t \geq 0}$ be a Hunt process with transition semigroup $(P_t)_{t \geq 0}$. We define for $\alpha >0$ and a bounded Borel-function $f$ the potential $U^{\alpha}f(x)=\int^{\infty}_0 e^{-\alpha t} P_t f(x) dt$. Why is the mapping $t \rightarrow e^{-\alpha t}U^{\alpha}f(X_t)$ right continuous?
In the case that $f \geq 0$ I already showed that $(e^{-\alpha t}U^{\alpha}f(X_t))$ is a supermartingale and therefore the mapping $t \rightarrow e^{-\alpha t}U^{\alpha}f(X_t)$ is right continuous.
How can I show this for the more generell case of a bounded Borel function?