Hello!

Could you please help me with this question:

Prove the following (known as Pratt's Lemma):

Let $\displaystyle \{f_k\}_{k\geq 1} , \{g_k\}_{k\geq 1} , \{G_k\}_{k\geq 1}$ be sequences of integrable functions. If

1) $\displaystyle f_k (x) \rightarrow f(x), g_k (x) \rightarrow g(x), G_k (x) \rightarrow G(x),$ as $\displaystyle k \rightarrow \infty \forall x $

2) $\displaystyle g_k (x) \leq f_k (x) \leq G_k (x) $ $\displaystyle \forall k \in \mathbb{N} $ $\displaystyle \forall x $

3) $\displaystyle \int g_k (x) dx \rightarrow \int g(x) dx $ and $\displaystyle \int G_k (x) dx \rightarrow \int G(x) dk $ as $\displaystyle k \rightarrow \infty $ with $\displaystyle \int g(x) dx$ and $\displaystyle \int G(x) dx $ both finite,

then

$\displaystyle lim_{k \rightarrow \infty} \int f_k (x) dx = \int f(x) dx$ and $\displaystyle \int f(x) dx $ is finite.

I was thinking about mimicking the proof of the Dominated Convergence Theorem (noting that $\displaystyle f_k (x) - g_k (x) \geq 0 $ and by Fatou's Lemma, concluding that $\displaystyle \int (f_k-g)(x) \leq \liminf_{n \rightarrow \infty} \int (f_k - g_n) (x) dx $ ) but I'm not quite sure where to go from there... Any hints/help is appreciated