. Let $T$ and $T'$ be two arbitrary stopping times satisfying $0\leq T<T'<\infty$ (a.s.). The discounted reward accumulated over the interval $T\leq t\leq T'$ is given by the random variable $$R_{T}^{T'}:= R(X_T,a_{T+0}, X_{T+0}) + \sum_{t=T+0}^{T'-0}\left[ \prod_{\tau=T}^{t-0}\Gamma\bigl(X_{\tau},a_{\tau+0}, X_{\tau+0}\bigr)\right] R\bigl(X_t,a_{t+0}, X_{t+0}\bigr)$$ where $a_{t+0}\sim\pi_{\ensuremath{\mathsf{c}}\xspace}(X_t)$ for $t=T,\ldots,T'-0$, and we set $R_T^T \equiv 0$ for any $T$.  