Book keeping.
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\section{Approximations with Simple Functions}
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\label{section:simple-approx}
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\begin{definition}[Admissible Approximant Function]
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\begin{definition}[Admissible Approximant Function*]
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\label{definition:admissible-approximant-function}
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Let $X$ be a topological space and $\mathcal{A}: X \to 2^X$, then $\mathcal{A}$ is an \textbf{admissible approximant function} on $X$ if:
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\begin{enumerate}[label=(AA\arabic*)]
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\end{lemma}
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\begin{definition}[Approximation of the Identity]
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\begin{definition}[Approximation of the Identity*]
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\label{definition:approximation-id-measure}
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Let $X$ be a topological space and $\net{I} \subset X^X$ be a net, then $\net{I}$ is an \textbf{approximation of the identity} if:
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\begin{enumerate}[label=(AI\arabic*)]
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