Let $(X_t)_{t \in [0,1]}$ be a semimartingale such that $E[|X_t|] < \infty$ for all $t\in [0,1]$. Let $f:[0,1] \rightarrow \mathbb R$ be a real function such that $g(t)=\int_0^t f(s) d X_s$ is finite for all $t\in [0,1]$.
Under which necessary and sufficient conditions does $E[g(t)]= \int_0^t f(s) d E[X_s]$ ? I am tempted by saying that this equality holds always. But what if $E[X_s]$ is not of bounded variation?
Related question I asked a few days ago: Is the mean of an integrable semimartingale of bounded variation?