Simple Functions are Dense in the Lebesgue Space

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space, let $p$ be a real number with $1\le p$, write $\mathcal{L}^{p}$ for the set of $p$-integrable functions on $(X,\mathcal{F},\mu)$ and $\lVert\cdot\rVert_{p}$ for the $p$-seminorm, and let $L^{p}=L^{p}(X,\mathcal{F},\mu)$ be the Lebesgue space, a real normed space by The Riesz–Fischer Theorem (i). Then the following hold.

1. (Which simple functions are $p$-integrable)(i) Let $s$ be a simple function on $(X,\mathcal{F})$. Then $s\in\mathcal{L}^{p}$ if and only if the set $\{x\in X:s(x)\ne 0\}$ belongs to $\mathcal{F}$ and has finite measure.

2. (Approximation)(ii) Let $f\in\mathcal{L}^{p}$ and let $\varepsilon$ be a positive real number. Then there is a simple function $s\in\mathcal{L}^{p}$ with $\lVert f-s\rVert_{p}\le\varepsilon$.

3. (Density)(iii) The set

$$S=\bigl\{[s]\ :\ s\text{ a simple function on }(X,\mathcal{F})\text{ with }s\in\mathcal{L}^{p}\bigr\}$$

is dense in $L^{p}$.