My book of stochastic analysis, in a proof, says:
The subset $H_t$ is measurable in $[0, \infty) \times C$, because $(s,f) \rightarrow f(t-s)$ is continuos in $[0,t] \times C$
$C$ is the space of continuos functions $C([0,\infty),\mathbb{R})$ and $H_t$ is $\left \{ (s,f) \in [0, \infty) \times C([0,\infty),\mathbb{R}) : s \leq t \textrm{ and } f(t-s) \le 0\right \}$.
I'm convinced this is so, but I can not prove it. The result is valid only in $[0,t] \times C$ or it is valid in $[0, \infty) \times C$ if I remove the hypothesis $s \leq t$ and extend $f \in C$ to $f(x) = 0 \textrm{ if }x > t$?