We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \in \mathbb T}, \mathbb P)$. Let $g : \mathbb T \times \mathbb R^d \to \mathbb R$ be continuous. Assume that $$ \int_0^t g(s, X_s) \, \mathrm d s $$ is well-defined and $\mathbb P$-integrable for each $t \in \mathbb T$.

Can we interchange the deterministic and stochastic integrals, i.e., $$ \mathbb E \left [ \int_0^t g(s, X_s) \, \mathrm d s \right ] = \int_0^t \mathbb E \left [ g(s, X_s) \right ] \mathrm d s $$ ?

Thank you so much for your elaboration!