Characterization of Stopping Times with respect to Right-Limit Filtration

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.

Let $T : \Omega \to \closedint 0 \infty$ be a random variable.

Let $\sequence {\GG_t}_{t \ge 0}$ be the right-limit filtration associated with $\sequence {\FF_t}_{t \ge 0}$.


 * $(1) \quad$ $T$ is a stopping time with respect to $\sequence {\GG_t}_{t \ge 0}$
 * $(2) \quad$ for each $t \in \hointr 0 \infty$, we have $\set {\omega \in \Omega : \map T \omega < t} \in \FF_t$
 * $(3) \quad$ for each $t \in \hointr 0 \infty$, the pointwise minimum $T \wedge t$ is $\FF_t$-measurable.