Exercise 1.4.47 (Defect version of Fatou's lemma)

$$
We have to demonstrate that

• From the pointwise dominance $f_n-f\le |f-f_n|$ using the monotonicity of the integral we obtain

  $$\left(f_n-f\right)\le \left|f-f_n\right|$$

  for each $n\in \mathbb{N}$. Taking the limits preserves the inequality.

• The first key observation we must make here is that the absolute value function $|f-f_n|$ can be pointwise repsented using $\min <!--\; FUNCTION\; APPLICATION\; -->,\max <!--\; FUNCTION\; APPLICATION\; -->$ functions: $|f-f_n|=\max <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)-\min <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)$ since both $f$ and $f_n$ are non-negative. Both $\max <!--\; FUNCTION\; APPLICATION\; -->$ and $\min <!--\; FUNCTION\; APPLICATION\; -->$ are measurable and absolutely integrable; thus, the theorem assertion reduces to

  $$\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{f}_n-{\int \; <!--\; nolimits\; -->}_{X}fd\mu \ge \underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\max <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)+\min <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)$$

  The second observation is that $\min <!--\; FUNCTION\; APPLICATION\; -->{(f,f_n)}_{n\in \mathbb{N}}\to f$ is a sequence of measurable functions dominated by the absolutely integrable function $f$; thus, by Dominated Convergence Theorem, $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\min <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)=\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\min <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)={\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{X}fd\mu$. Cancelling this term out, we further reduce the theorem assertion to

  $$\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{f}_n\ge \underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\max <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)$$

  Unfortunately we cannot apply the same trick for $\max <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)$ since it is not necessarily dominated. We can, however, apply Fatou’s lemma (Corollary 1.4.46) to get

  $$\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{f}_n-\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\max <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)\ge \left[\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{f}_n-\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\max <!--\; FUNCTION\; APPLICATION\; -->(f,f_n)\right]=f-f=0$$