Given a joint measurable, adapted process $f(t,\omega)$ with $\int_a^b|f(t,\omega)|^2 dt< \infty$ almost surely. The expectation $E[\int_a^b|f(t)|^2dt]$ may not exist. However, given a process $\int_a^b|f(t,\omega)|^2dt<n$, one can use the monotony of the expectation to conclude $E[\int_a^b|f(t)|^2dt]<E[n]=n$ and using Fubini $E[\int_a^b|f(t)|^2dt]=\int_a^bE[|f(t)|^2]dt<n$.
Now my question is, what is the distinction between two cases. I mean from $\int_a^b|f(t,\omega)|^2 dt< \infty$ I can conclude there exist an $n$ such that $\int_a^b|f(t,\omega)|^2dt<n$, which make the both cases equivalent. Why I can then define the expectation value in the second case and not in the first?
Thank you
The distinction lies between the integral being finite and it being bounded. $X < \infty$ $, \mathbb{P}$-a.s. does not imply that $\exists n \mathrm{\ s.t.\ } X < n$. Consider the function $X(\omega)=e^{w^3}$ which is finite a.s., i.e., $\mathbb{P}(\omega \in \Omega:X< \infty)=1$ (for the Gaussian measure on $\mathbb{R}$). This doesn't imply that it is integrable.
Also, the theorem you would use (in the bounded case) would be Tonelli's not Fubini's. I believe Fubini's assumes that the integral is defined over the product measure space, while Tonelli's requires only positivity of the integrand.