Expectation of a local martingale

Question

How do I prove that $\mathbb{E}[\int_o^t B^4_s(\omega) ds ]$ is finite for any $ t \geq 0$?

Here $B_t$ is the standard Brownian Motion.

If I can change the expectation with the integral the result follows. But, is there a theorem that allow me to do that?

Answer 1

$\int_0^{t} E|B_s|^{4} ds=c\int_0^{t} s^{2}ds<\infty$ where $c=EX^{4}$ and $X$ has standard normal distribution. [ This is because $B_s$ has same distribution as $\sqrt s X$]. No use Fubini/Tonelli Theorem to interchange the integral and the expectation to get $ E \int_0^{t}B_s^{4} ds<\infty$.