Lebesgue integral and a parametrized family of functions

Question

The main advantages of the Lebesgue integration are due to the theorems that "control" the (pointwise) limit of a sequence of measurable functions. In particular, I refer to the monotone convergence theorem, the Fatou's lemma and the dominated convergence theorem.

Now let $t\in\mathbb R$ be a real parameter; if $\{f_t\}$ is a family of real valued and measurable functions (clearly the index $t$ runs in an uncountable set), and we now that $$\displaystyle \lim_{t\to t_0} f_t(x)=f(x),$$ what about the quantity $\int f\,d\mu$? In this context are there some theorems like the monotone convergence theorem, the Fatou's lemma and the dominated convergence theorem? Under which hypothesis can I assert that $$\int f\,d\mu=\lim_{t\to t_0}\int f_t\,d\mu\;\;\;\textrm{?}$$

Answer 1

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 $\int f \ d\mu \ne \lim_{n \to \infty} \int f_{t_n} \ d\mu$.

Answer 2

$\lim_{x \to a} f(x) = A$ 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 $t \to t_0$.