This question is about the dominated convergence theorem for a stochastic process.
Let $X_t$ be a sequence of stochastic process on some probability space $(\Omega,\mathcal{F}_t,\mathbb{P})$. For each $\omega$ assume $X_\cdot(\omega)$ is continuous on $[0,T]$ (i.e the realisations of the stochastic process are continuous paths).
Say $X_t$ take values in $\mathbb{R}^d$. Let $f^n$ be a sequence of functions $f^n:\mathbb{R}^d \to \mathbb{R}$ converging point wise to some function $f$.
Consider an integral $$\lim_{n\to \infty}\int_0^T f^n(X_s)\,ds $$
Assume that for almost every $\omega$ there exists a function $g^\omega$ such that $f^n(X_s(\omega))\leq g^\omega(s)$, where $g^\omega$ is integrable, i.e $\int_0^T g^\omega(s)ds<\infty .$
Can one apply the dominated convergence theorem to say $$\lim_{n\to \infty}\int_0^T f^n(X_s)\,ds=\int_0^T\lim_{n\to \infty} f^n(X_s)\,ds \text{?} ~~~\text{almost surely}$$
What would be the case if there was a $g$ (independent of $\omega$) which did the dominating job for a.e $\omega$.