Approximation of Measurable Functions by Simple Functions

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space. Then the following hold.

1. (Nonnegative measurable functions)¶ Let $f:X\to[0,\infty]$ be measurable. Then there is a sequence $(s_{m})_{m\in\mathbb{N}}$ of nonnegative simple functions on $(X,\mathcal{F})$ such that

and such that $f(x)$ is the least upper bound of $\{s_{m}(x):m\in\mathbb{N}\}$ in $[0,\infty]$ for every $x\in X$.

2. (Bounded functions)¶ Let $K$ be a real number and let $f:X\to\mathbb{R}$ be measurable with $0\le f(x)\le K$ for every $x\in X$. Then there is a sequence $(s_{m})_{m\in\mathbb{N}}$ of nonnegative simple functions on $(X,\mathcal{F})$ with

3. (Real-valued measurable functions)¶ Let $f:X\to\mathbb{R}$ be measurable. Then there is a sequence $(s_{m})_{m\in\mathbb{N}}$ of simple functions on $(X,\mathcal{F})$ such that

and such that for every $x\in X$ the sequence $(s_{m}(x))_{m\in\mathbb{N}}$ converges to $f(x)$.