Let $X$ be a one-dimensional continuous-time Brownian Motion. Let $T=\inf\{t\geq0:X_t=1\}$, and define $H$ by $H_t=\mathbf{1}\{T\geq t\}$. Show that $H$ is a previsible process.
I think that $H$ is right-continuous so I can't just use the typical result that adapted left-continuous processes are previsible. I think that the previsibility of $H$ must come from the previsibility of the Brownian Motion $X$, but I'm not sure how to show this. Would greatly appreciate any help!