Conditions of Fubini's theorem

Question

I read already some answer about this matter but I'm still unsure. If i have $E[\int_0^tX_sds]$ where X is some stochastic process, and if I want to interchange expectation with integral, which are the conditions that i have to verify before applying Fubini?

Answer 1

If $X_s$ is measurable process in the sense $(s,\omega) \to X_s(\omega)$ is measurable on $[0,t]\times\Omega$ and if $\int_{\Omega \times [0,t] } |X_s(\omega)| d(P\times m) <\infty$ (where $m$ is the Lebesgue measuer) then we can apply Fubini's Theorem.