Let $\psi^n_t$ be a sequence of stochastic process. If I know that $\psi^n_t\to 0$ a.s. for all t.
Let $|\psi^n|< \phi$. And I also know that $E\bigl[\big| \int_0^t|\phi_s|ds\big|\bigr] <\infty$ and I want to prove that $\psi^n_t\to 0$ in $L^1$: so we have $$\lim_{n\to \infty} E\bigg[\int_0^t\psi_s^nds \bigg]= E\bigg[\lim_{n\to \infty}\int_0^t\psi_s^nds \bigg]$$ follows from the dominated convergence theorem. But now, how can I bring the limit inside the Lebesgue integral? I know that $\psi^n< \phi$ but it seems to me that I don't know if $\phi \in L^1$ for applying Lebesgue dominated theorem again.
You need $|\psi^{n}| <|\psi|$ for this. If you have this then you are just applying DCT to the product space $(\Omega,\mathcal F, P) \times ((0,t), \mathcal B, m)$ where $m$ is Lebesgue measure.