Simple function under-approximation to a numerical function

See notes page: Notes:Simple function approximation to a numerical function

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Important for measure theory, and needs a name. SNAF [ilmath]\leftarrow[/ilmath] simple numerical approximation function

[ilmath]\newcommand{snaf}{\text{Snaf} } [/ilmath]

Definition

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

EARLY VERSION

Definition

TODO: I need to come up with better definitions

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

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

Reference

<reference/>