I'm reading a book "Introduction to random process" written by N. V. Krylov. I have a question a fact which the author does not give a proof and any references.
I have no idea to prove it or find a relevant references.
$\mathcal{S}$ denote the set of all $\mathcal{F}_t$-adapted, $\mathcal{F}\otimes \mathcal{B}(0,\infty)$-measurable process $f_t$ such that $$ \int_0^t f_s^2 ds<\infty \quad\text{a.s. }\quad T<\infty.$$
By $H$ we denote the set of all real-valued $\mathcal{F}_t$-adapted functions $f_t(\omega)$ which are $\mathcal{F}\otimes \mathcal{B}(0,\infty)$-measurable and satisfy $$ \mathbb{E}\int_0^\infty f_t^2 dt<\infty. $$
The author gives the following fact without giving a proof and any reference.
If $f\in\mathcal{S}$, $\tau$ is a stopping time, and $$ \mathbb{E} \int_0^\tau f_s^2 ds = \mathbb{E} \int_0^\infty I_{s<\tau} f_s^2 ds<\infty, $$ then $I_{s<\tau} f_s \in H$ and $\int_0^\infty I_{s<\tau} f_s dw_s$ make sense.
How can I prove this fact? Thanks in advanced.