Exercise 1.4.46 (Generalised dominated convergence theorem)

Without the loss of generality we make the following assumptions:

• By modifying $f_n$ and $f$ on a null set, we may assume without loss of generality that $f_n$ converge to $f$ pointwise everywhere.
• Similarly, we may assume that $\left|f_n\right|$ are bounded by $G+g_n$ everywhere.
• Since we can always split the complex integral into the real and the imaginary part, we only look at the case where $f_n,f$ are real. The assertion for the complex valued integrals then follows easily by additivity.

Since $\int g_n\to 0$ we conclude $g_n\to 0$.

We now demonstrate both sides of inequality

• From $\left|f_n\right|\le g_n+G$, i.e., from $-g_n-G\le f_n\le G+g_n$ we see that the function $0\le f_n+g_n+G$ is non-negative. This allows us to apply Fatou’s lemma (Corollary 1.4.46) and conclude

  $$\left(f+G\right)\le \underset{n\to \infty }{\mathrm{liminf}<!--\; FUNCTION\; APPLICATION\; -->}\left(f_n+g_n+G\right)$$

  or, in other words,

  $${\int \; <!--\; nolimits\; -->}_{X}fd\mu +G\le \underset{n\to \infty }{\mathrm{liminf}<!--\; FUNCTION\; APPLICATION\; -->}{f}_n+\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{g}_n+G$$

  Removing the vanishing integral of $g_n$ and subtracting the finite quantity $\int G$ from both sides we obtain the desired result

  $${\int \; <!--\; nolimits\; -->}_{X}fd\mu \le \underset{n\to \infty }{\mathrm{liminf}<!--\; FUNCTION\; APPLICATION\; -->}{f}_n$$

• Similarly, from $\left|f_n\right|\le g_n+G$, i.e., from $-g_n-G\le f_n\le G+g_n$ we see that the function $0\le G+g_n-f_n$ is non-negative. Applying Fatou’s lemma, we get

  $$\left(G-f\right)\le \underset{n\to \infty }{\mathrm{liminf}<!--\; FUNCTION\; APPLICATION\; -->}\left(G+g_n-f\right)\le \underset{n\to \infty }{\mathrm{limsup}<!--\; FUNCTION\; APPLICATION\; -->}\left(G+g_n-f\right)$$

  or, in other words,

  $$G-{\int \; <!--\; nolimits\; -->}_{X}fd\mu \le G+\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{g}_n-\underset{n\to \infty }{\mathrm{limsup}<!--\; FUNCTION\; APPLICATION\; -->}{f}_n$$

  Removing the vanishing integral of $g_n$ and subtracting the finite quantity $\int G$ from both sides and negating the inequality we obtain

  $$\underset{n\to \infty }{\mathrm{liminf}<!--\; FUNCTION\; APPLICATION\; -->}{f}_n\le {\int \; <!--\; nolimits\; -->}_{X}fd\mu$$

  as desired.