Lebesgue integral and a parametrized family of functions

Monotone convergence, Fatou's lemma and dominated convergence should all apply without change. Note that if $\int f \ d\mu \ne \lim_{t \to t_0} \int f_t \ d\mu$, there is a monotone sequence $t_n \to t_0$ such that $\int f \ d\mu \ne \lim_{n \to \infty} \int f_{t_n} \ d\mu$.

$\lim_{x \to a} f(x) = A$ if and only if for every sequence $\{x_n\}$ that converges to $a$, we have $\lim_{n \to \infty} f(x_n) = A$.

The convergence theorems hold for every sequence $\{t_n\}$ that converges to $t_0$. Thus, they also hold when we take the limit $t \to t_0$.