Let $(X^n_t)_{t \geq 0}$ be a sequence of random real-valued processes that converges in finite-dimensional distributions, i.e. for all $k \in \mathbb{N}$ and for all $0 \leq t_1 < \dots < t_k$ the distribution of $(X^n_{t_1}, \dots, X^n_{t_k})$ converges weakly to some probability measure on $\mathbb{R}^k$ as $n \to \infty$.
Is it then true that the process $\int_0^t X^n_s \, ds$ converges in finite-dimensional distributions? (Assume that $X^n_s$ is integrable, e.g. cadlag).
The most important step seems to be to check that this holds for simple processes, e.g. $X^n_s$ taking values in $\{ 1, 2 \}$. Then how can I show that $P( |\{ s \in [0,t] \ | \ X^n_s = 1 \}| \leq x)$ converges, where $|.|$ denotes the Lebesgue measure? Or equivalently, why do such random sets converge in distribution?