measurability of stopped processes

Let $X$ be a real-valued stochastic process and $\tau$ be a stopping time. If $X$ satisfies any of the following properties then so does the stopped process $X^{\tau }$.

1. $X$ is jointly measurable.

2. $X$ is progressively measurable.

3. $X$ is optional.

4. $X$ is predictable.

In particular, if $X$ is a right-continuous and adapted process then it is progressive (alternatively, it is optional). Then, the stopped process $X^{\tau }$ will also be progressive and is therefore right-continuous and adapted.

Also, for any progressive process $X$ and bounded stopping time $\tau \le t$, the above result shows that $X_{\tau }=X_t^{\tau }$ will be ${ℱ}_t$-measurable.