I solve all the exercises on Page 52. However, I’m too lazy to write all the solutions. I’ll just post the last two.
$\textbf{Problem 16.}$ If $f \in L^{+}$and $\int f<\infty$, for every $\epsilon>0$ there exists $E \in \mathcal{M}$ such that $\mu(E)<\infty$ and $\int_E f>\left(\int f\right)-\epsilon$.
$\textbf{Solution.}$ Let $E_n:=\{x: f(x)\ge \frac{1}{n}\}$. It is obvious that $f1_{E_n}$ is an increasing function such that $\lim f_n =f$. Therefore, by the MCT we have that $$\lim \int f_n = \int_{En} f = \int f$$. Thefore, there is an $n$ such that $$\int f_n > \int f -\epsilon$$. Interestingly, $\mu(E_n)<\infty$ because $\int f < \infty$. One can show that the set ${x:f(x)>0}$ is $\sigma$-finite.
$\textbf{ Problem 17.}$ Assume Fatou’s lemma and deduce the monotone convergence theorem from it.
$\textbf{Solution.}$ Suppose $\{f_n\} \subset L^+$ and increasing such that $$f_n \rightarrow f$$. By Fatou’s, we have that
$$ \int \liminf f_n = \int f \le \liminf \int f_n$$.
To prove the other inequality, notice that $f_n \le f$ so that $ \int f_n < \int f$. This implies that $$ \liminf \int f_n \le \int f$$.