Simple function under-approximation to a numerical function

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Important for measure theory, and needs a name. SNAF [ilmath]\leftarrow[/ilmath] simple numerical approximation function
[ilmath]\newcommand{snaf}{\text{Snaf} } [/ilmath]


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



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][Note 1] 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.

Construction of [ilmath]s[/ilmath]


  1. if [ilmath]a:X\rightarrow\overline{\mathbb{R} } [/ilmath] is our approximating function, then to be an under-estimation:
    • [ilmath]\forall x\in X[a(x)\le f(x)][/ilmath]
    Note that this would be an "over-estimation" (sort of) in the negative case. This is one of the reasons we forbid [ilmath]f[/ilmath] from ever being below zero