If $X$ is a stochastic process, a.s. continuos and such that $\forall t \geq 0, X_t \in L^1_\omega$, is its mean function $t \rightarrow E[X_t]$ continuos?
I can show it if $X \in L^1_\omega L^{\infty}_t$ which means that $E[\sup_{t \geq0} |X_u|] < \infty$ by the dominated convergence theorem.
Is it true in general?