Proof of Simple Functions are Dense in the Lebesgue Space

Each result cited is universally quantified over the data appearing in its own statement, and is applied here to the data named in the statement above. We use countable choice, to select one approximating simple function for each index.

Claim 1. If $X$ is empty then $Z=\varnothing$ has measure $0$ and the unique map $X\to\mathbb{R}$ lies in $\mathcal{L}^{p}$, its integral being $0$; both sides of the asserted equivalence hold and there is nothing to prove. Assume therefore that $X$ is nonempty, so that a simple function on $(X,\mathcal{F})$ has at least one value.

Let $s=\sum_{i=1}^{r}c_{i}\mathbf{1}_{A_{i}}$ be the standard representation of Simple Functions, Nonnegative Simple Functions, and the Simple Integral, with the $c_{i}$ the distinct values of $s$ and $A_{i}=s^{-1}(\{c_{i}\})\in\mathcal{F}$. The set $Z=\{x\in X:s(x)\ne0\}$ is the union of those $A_{i}$ with $c_{i}\ne0$, a union of finitely many members of $\mathcal{F}$, so $Z\in\mathcal{F}$ in every case; only its measure is at issue.

Suppose $\mu(Z)<\infty$. Let $K$ be the largest of the finitely many numbers $|c_{1}|,\dots,|c_{r}|$. For $x\notin Z$ we have $s(x)=0$ and hence $(|s(x)|)^{p}=0$, while for $x\in Z$ we have $|s(x)|\le K$ and hence $(|s(x)|)^{p}\le K^{p}$ by Properties of Real Powers (v). So $|s|^{p}\le K^{p}\mathbf{1}_{Z}$ pointwise, and by the monotonicity and homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion together with The Integral of an Indicator Function is the Measure of the Set,

$$\int_{X}|s|^{p}\,d\mu\le K^{p}\mu(Z)<\infty ,$$

so $s\in\mathcal{L}^{p}$, the map $s$ being measurable.

Conversely suppose $s\in\mathcal{L}^{p}$. If $Z$ is empty then $\mu(Z)=0<\infty$. Otherwise let $c$ be the smallest of the numbers $|c_{i}|$ with $c_{i}\ne0$, a positive real number, so that $c\le|s(x)|$ for every $x\in Z$ and hence $c^{p}\mathbf{1}_{Z}\le|s|^{p}$ pointwise by Properties of Real Powers (v). As above this gives $c^{p}\mu(Z)\le\int_{X}|s|^{p}\,d\mu<\infty$. Since $c^{p}$ is positive by Properties of Real Powers (i), the value $\mu(Z)=\infty$ would force $c^{p}\mu(Z)=\infty$; hence $\mu(Z)<\infty$.

Claim 2. By Approximation of Measurable Functions by Dyadic Simple Functions (iii) there is a sequence $(s_{m})_{m\in\mathbb{N}}$ of simple functions with $|s_{m}|\le|f|$ pointwise and with $(s_{m}(x))_{m}$ converging to $f(x)$ for every $x\in X$. By Elementary Properties of the p-Seminorm (iii) each $s_{m}$ lies in $\mathcal{L}^{p}$.

For every $x$ and $m$, claims 2 and 5 of Properties of the Absolute Value in an Ordered Field give $|f(x)-s_{m}(x)|\le|f(x)|+|s_{m}(x)|\le 2|f(x)|$, so by Properties of Real Powers (v) and Properties of Real Powers (iii),

$$\bigl(|f(x)-s_{m}(x)|\bigr)^{p}\le 2^{p}\bigl(|f(x)|\bigr)^{p},$$

and the map $2^{p}|f|^{p}$ is integrable because $f\in\mathcal{L}^{p}$. Moreover $(|f(x)-s_{m}(x)|)_{m}$ converges to $0$ for every $x$, hence so does $((|f(x)-s_{m}(x)|)^{p})_{m}$ by Properties of Real Powers (vii) together with $0^{p}=0$. The maps $|f-s_{m}|^{p}$ are measurable by Power-Integrable Functions and the p-Seminorm (i), so The Lebesgue Integral and Null Sets (vii), applied with the limit function $0_{X}$, gives

$$\lim_{m\to\infty}\int_{X}|f-s_{m}|^{p}\,d\mu=0 .$$

By Elementary Properties of the p-Seminorm (i) this integral is $(\lVert f-s_{m}\rVert_{p})^{p}$, and applying Properties of Real Powers (vii) with the exponent $1/p$, together with $((\lVert f-s_{m}\rVert_{p})^{p})^{1/p}=\lVert f-s_{m}\rVert_{p}$ and $0^{1/p}=0$, we get that $\lVert f-s_{m}\rVert_{p}$ converges to $0$. In particular there is $m$ with $\lVert f-s_{m}\rVert_{p}\le\varepsilon$, and $s=s_{m}$ is as required.

Claim 3. Let $F\in L^{p}$ and let $f\in\mathcal{L}^{p}$ be a representative of $F$. For each $m\in\mathbb{N}$ apply claim 2 with the positive number $1/m$ and choose a simple $s_{m}\in\mathcal{L}^{p}$ with $\lVert f-s_{m}\rVert_{p}\le 1/m$. Then $[s_{m}]\in S$ and

$$d_{p}\bigl(F,[s_{m}]\bigr)=\lVert f-s_{m}\rVert_{p}\le\frac{1}{m},$$

by the formula for $d_{p}$ in The Riesz–Fischer Theorem (i). Since $(1/m)_{m}$ converges to $0$, claim 3 of Order Properties of Limits of Real Sequences shows that $(d_{p}(F,[s_{m}]))_{m}$ converges to $0$, that is, $([s_{m}])_{m}$ converges to $F$ in $(L^{p},d_{p})$. By Sequential Characterization of the Closure in a Metric Space the point $F$ lies in the closure of $S$. As $F$ was arbitrary, that closure is all of $L^{p}$, so $S$ is dense in $L^{p}$.