For the continuous time case, is there any example such that $\tau_1, ..., \tau_n$ are stopping times, but $\inf_n \tau_n $ is not a stopping time?

Question

For the continuous time case, is there any example such that $\tau_1, ..., \tau_n$ are stopping times, but $\inf_n \tau_n $ is not a stopping time?

We know if the filtration is right continuous, then $\inf_n \tau_n $ is always a stopping time, so the filtration in such an example must be non-right continuous.

Answer 1

Here is an example for K.L. Chung's book Lectures from Markov Processes to Brownian Motion. Consider a Poisson process with left continuous paths, and let $T$ be the time of the first jump. Then $T_n=T+1/n$ is a stopping time for each $n$, but $T=\inf_n T_n$ itself is not.