I have a question concerning the dominated convergence theorem. For two sequences $(X_n)_{n\in\mathbb{N}}$, $(Y_n)_{n\in\mathbb{N}}$ of random variables and a stopping time $\tau$ it is said that:
$\mathbb{E}[F_1(X_n)F_2(Y_n)\mathbf{1}_{\{\tau>n\}}]$=$\mathbb{E}[F_1(X_n)F_2(Y_n)\mathbf{1}_{\{\tau>n,|X_n|\geq n^{\varepsilon}\}}]+\sigma_n(1)$
for two bounded measurable functions $F_1$ and $F_2$ with the dominated convergence theorem for $\varepsilon\in(0,1)$.
Have someone a tip for me how the dominated convergence theorem is used there?
Thank you in advance!