How to formally proof this measurability?

Question

I have

$Y = \int_0^t X_s ds$

It is intuitive that $Y$ is $\{\sigma(X_s), s \le t, \}$-measurable, since it onl y depends on the values of the process $X_s$ up until $X_t$ but how to write it formally ?

Answer 1

Let $\mathcal F_t$ denote $\sigma\{X_s:0\le s\le t\}$ and let $\mathcal B_{0,t]}$ denote the Borel subsets of $[0,t]$. If $X$ is bounded or positive, and if $\Omega\times[0,t]\ni(\omega,s)\mapsto X_s(\omega)$ is $\mathcal F_t\otimes\mathcal B_{[0,t]}$-measurable, then $Y$ will be $\mathcal F_t$-measurable, by Fubini's Theorem.