While reading up on "Glivenko Cantelli Theorem" from Probability Models by K.B Athreya, the author used 2 lemmae to prove it. One was called Scheffe's lemma, the other Polya's theorem.
Scheffe's Lemma is stated as follows:
Let $f_n, f$ be non negative $\mu$ integrable functions. If $f_n \to f$ a.e and $\int f_n d\mu \to \int fd\mu$, then
$$\int |f_n – f|d\mu \to 0$$
My proof is:
Let $g_n = |f_n -f|$. Now we have $g_n \to 0$ a.e. Now
$$0 \leq g_n = |f_n -f| \leq f + f_n$$
$$\Rightarrow \int g_n d\mu \leq \int fd\mu + \int f_nd\mu < \infty $$
Thus by Dominated convergence theorem,
$$\int g_nd\mu \to 0$$ QED.
The Question:
My doubt is that in this proof, I have not used that $\int f_n d\mu \to \int fd\mu$, at least not explicitly. So is this condition superfluous?
What I searched:
I searched for Scheffe on MSE, but got 3 results (none useful) and when I typed Scheffe's instead, I got a result not belonging to these 3 which was actually on Scheffe's lemma (Though not helpful). It's strange (not the result but the search).
I'd appreciate any help/hints on this. Kindly ask me for clarifications if required.
Best Answer
Let
$$u_n = \max\{ f, f_n\} \quad \text{and} \quad l_n = \min \{ f, f_n \} $$
so that both $(u_n)$ and $(l_n)$ converge pointwise to $f$, $l_n \leq f \leq u_n$ and $|f - f_n| = u_n - l_n$. By DCT, it is clear that $\int l_n \, d\mu \to \int f \, d\mu$. thus from
$$ \int u_n \, d\mu = \int (f + f_n - l_n) \, d\mu ,$$
taking $n\to\infty$, we have $\int u_n \, d\mu \to \int f \, d\mu$. (Here the assumption that you are concerning is used.) Therefore we have
$$ \int |f - f_n| \, d\mu = \int u_n \, d\mu - \int l_n \, d\mu \to 0. $$