Simple function under-approximation to a numerical function

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Important for measure theory, and needs a name. SNAF $\leftarrow$ simple numerical approximation function

$\newcommand{snaf}{\text{Snaf} }$

Definition

Let $(X,\mathcal{A})$ be a measurable space. A $\Snaf$ is a simple numerical approximation function

EARLY VERSION

Definition

TODO: I need to come up with better definitions

Let $(X,\mathcal{A})$ be a measurable space and let $f:X\rightarrow\overline{\mathbb{R} }$ be an $\mathcal{A} /$$\mathcal{B}(\overline{\mathbb{R} })$-measurable function that is non-negative, i.e. $\forall x\in X[f(x)\ge 0]$, then we can construct a (non-negative) simple function that (under)-approximates $f$ as follows:

• $\text{Snaf}:\mathbb{N}_{\ge 1}\times\mathbb{R}_{\ge 0}\rightarrow\mathcal{E}(\mathcal{A})$ - recall that $\mathcal{E}(\mathcal{A})$ denotes the set of all simple functions on $\mathcal{A}$ and that simple functions by their nature have the reals as their co-domain.
  • We could say the mapping $\text{Snaf}$ is given by: $\text{Snaf}:(n,r)\mapsto (s:X\rightarrow\mathbb{R})$, we construct $s$ below.

Reference

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