I have been reading about stopping times and have a question about the composition of stopping times with a measurable function.
Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be any measurable function with $f(t)\geq t$. If $\tau$ is a stopping time, is $f(\tau)$ also a stopping time?
In his blog, George Lowther says this should be a stopping time. However, I am not sure why this is. Well, clearly, $f(\tau)$ is a well-defined measurable function and we have $\{f(\tau)\leq t\}\subseteq \{\tau\leq t\}\in \mathcal{F}_t$. However, we still need to show that $\{f(\tau)\leq t\}\in\mathcal{F}_t$. I am unsure about how to show this. Any help would be appreciated.
$\{f(\tau) \leq t\}=\{\tau \in f^{-1}(-\infty, t]\}$ and $f^{-1}(-\infty, t] \subseteq (-\infty, t] $ because of the hypothesis on $f$.
Hint for the rest of the argument: Consider the collection of all Borel sets $A$ in $(-\infty, t]$ such that $\tau^{-1}(A) \in \mathcal F_t$. This is a $\sigma-$ field and is contains all intervals of the type $(a,b]$ contained in $(-\infty, t]$.