View source for Simple function under-approximation to a numerical function

: '''See notes page: ''' [[Notes:Simple function approximation to a numerical function]]

{{Stub page|grade=A|msg=Important for measure theory, and needs a name. SNAF {{M|\leftarrow}} ''simple numerical approximation function''}}{{M|\newcommand{snaf}{\text{Snaf} } }}
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==Definition==
Let {{M|(X,\mathcal{A})}} be a [[measurable space]]. A {{M|\Snaf}} is a ''simple numerical approximation function''

=EARLY VERSION=
==Definition==
: {{XXX|I need to come up with better definitions}}
Let {{M|(X,\mathcal{A})}} be a [[measurable space]] and let {{M|f:X\rightarrow\overline{\mathbb{R} } }} be an {{M|\mathcal{A} / }}[[Borel sigma-algebra of the extended real line|{{M|\mathcal{B}(\overline{\mathbb{R} })}}]]-''[[measurable function]]'' that is ''[[non-negative]]'', {{ie}} {{M|\forall x\in X[f(x)\ge 0]}}, then we can construct a (non-negative) {{link|simple function|measure theory}} that (under)-approximates {{M|f}}<ref group="Note">if {{M|a:X\rightarrow\overline{\mathbb{R} } }} is our approximating function, then to be an ''under-estimation'':
* {{M|\forall x\in X[a(x)\le f(x)]}}
Note that this would be an "over-estimation" (sort of) in the negative case. This is one of the reasons we forbid {{M|f}} from ever being below zero</ref> as follows:
* {{M|\text{Snaf}:\mathbb{N}_{\ge 1}\times\mathbb{R}_{\ge 0}\rightarrow\mathcal{E}(\mathcal{A})}} - recall that {{M|\mathcal{E}(\mathcal{A})}} denotes the [[set of all simple functions]] on {{M|\mathcal{A} }} and that simple functions by their nature have [[the reals]] as their {{link|co-domain|function}}.
** We could say the [[mapping]] {{M|\text{Snaf} }} is given by: {{M|\text{Snaf}:(n,r)\mapsto (s:X\rightarrow\mathbb{R})}}, we construct {{M|s}} below.
===Construction of {{M|s}}===

==Notes==
<references group="Note"/>
==Reference==
<reference/>
{{Definition|Measure Theory}}