Exercise 1.3.11 (Bounded Lebesgue measurable functions are Lebesgue integrable)

Let $S$ be the finite measure support of $f$. The key to this proof is to use Exercise 1.3.4, which states that we can find a sequence ${(g_n)}_{n\in \mathbb{N}}$ which is (1) increasing $g_1\le g_2\le \dots$, (2) has finite measure support $S$ and (3) converges uniformly. Note that we can cherry-pick a subsequence of our original sequence such that

Furthermore we can easily construct a sequence ${(h_n)}_{n\in \mathbb{N}}$ of simple functions which majorizes $f$ and also converges uniformly by setting $h_n:=g_n+\frac{2}{n}$. We then have $h_n-f=g_n+2∕n-f\ge 1∕n>0$ and $d_{\infty }(h_n,f)\le d_{\infty }(h_n,g_n)+d_{\infty }(g_n,f)\le \frac{2}{n}+\frac{1}{n}$. We now demonstrate that by converging to each other

these sequences also drag the lower and upper Lebesgue integrals towards each other.

To do so, we demonstrate that $\mathrm{Simp}{\int }_{{ℝ}^d}{g}_n$ can get arbitrarily close to $\underline{{\int }_{{ℝ}^d}}f$. By definition, we can pick a simple function $g^{\prime }\le f$ such that $\underline{{\int }_{{ℝ}^d}}f-\mathrm{Simp}{\int }_{{ℝ}^d}{g}^{\prime }\le 1∕n$. We then have

This quantity can get arbitrarily small as $n\to \infty$, so we conclude $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\mathrm{Simp}{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{{ℝ}^d}{g}_n=\underline{{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{{ℝ}^d}}f$. Similarly, for some majorizing function $h^{\prime }\ge f$ with $\mathrm{Simp}{\int }_{{ℝ}^d}{h}^{\prime }-\underline{{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}}f\le \frac{1}{n}$ we have

And hence $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}\mathrm{Simp}{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{{ℝ}^d}{h}_n=\overline{{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_{{ℝ}^d}}f$. The conclusion is