185 lines
9.6 KiB
TeX
185 lines
9.6 KiB
TeX
\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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\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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\item For each $x \in X$, $x \in \overline{\mathcal{A}(x)^o}$.
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\item $\bigcap_{x \in X}\mathcal{A}(x) \ne \emptyset$.
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\end{enumerate}
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and $\mathcal{A}$ is \textbf{Borel measurable} if:
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\begin{enumerate}[label=(AA\arabic*), start=2]
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\item[(B)] For any $x_0 \in X$, $\bracs{x \in X|x_0 \in \mathcal{A}(x)} \in \cb_X$.
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\end{enumerate}
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\end{definition}
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\begin{lemma}
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\label{lemma:admissible-approximant-existence}
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Let $X$ be a topological space, and $\mathcal{A}: X \to 2^X$ be defined by $x \mapsto X$, then $\mathcal{A}$ is an \hyperref[admissible approximant function]{definition:admissible-approximant-function}.
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\end{lemma}
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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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\item For each $x \in X$, $I_\alpha(x) \to x$.
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\end{enumerate}
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For any \hyperref[admissible approximant function]{definition:admissible-approximant-function} $\mathcal{A}: X \to 2^X$, $\net{I}$ is \textbf{$\mathcal{A}$-admissible} if:
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\begin{enumerate}[label=(AI\arabic*), start=1]
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\item For each $x \in X$ and $\alpha \in A$, $I_\alpha(x) \in \mathcal{A}(x)$.
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\end{enumerate}
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The approximation $\net{I}$ is \textbf{simple} if $I_\alpha$ is finitely-valued for all $\alpha \in A$, and \textbf{Borel measurable} if $I_\alpha$ is Borel measurable for all $\alpha \in A$.
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\end{definition}
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\begin{lemma}[Existence of Simple Approximations of the Identity]
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\label{lemma:separable-metric-space-approx-identity}
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Let $X$ be a separable metric space, $\mathcal{A}: X \to 2^X$ be a Borel measurable \hyperref[admissible approximant function]{definition:admissible-approximant-function}, and $\seq{x_n} \subset X$ be a dense subset with $x_1 \in \bigcap_{x \in X}\mathcal{A}(x)$, then there exists $\seq{I_n} \subset X^X$ such that:
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\begin{enumerate}
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\item $\seq{I_n}$ is a Borel measurable, $\mathcal{A}$-admissible \hyperref[approximation of the identity]{definition:approximation-id-measure}.
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\item For each $N \in \natp$, $I_N(X) \subset \bracsn{x_n|1 \le n \le N}$.
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\item For each $N \in \natp$ and $x \in X$,
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\[
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d(x, I_N(x)) = \min\bracs{d(x, x_n)| 1 \le n \le N, x_n \in \mathcal{A}(x)}
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\]
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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By removing duplicate elements from the sequence, assume without loss of generality that for each $m, n \in \natp$ with $m \ne n$, $x_m \ne x_n$.
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Let $N \in \natp$. For each $x \in X$, let
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\[
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C_N(x) = \bracs{1 \le n \le N| x_n \in \mathcal{A}(x)}
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\]
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Since $x_1 \in \bigcap_{y \in X}\mathcal{A}(y)$, $1 \in C_N(x)$ and $C_N(x) \ne \emptyset$. Now, let
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\[
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k_N(x) = \min\bracs{n \in C_N(x) \bigg | d(x, x_n) = \min_{m \in C_N(x)}d(x, x_m)}
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\]
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be the minimum $n \in C_N(x)$ on which the minimal distance from $x$ to $\bracs{x_m|m \in C_N(x)}$ is achieved. Define
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\[
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I_N: X \to X \quad x \mapsto x_{k_N(x)}
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\]
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(2): For each $x \in X$, $k_N(x) \in [N]$, so $I_N(x) \in \bracsn{x_n|1 \le n \le N}$.
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(3): Let $x \in X$, then by definition of $k_N$ and $C_N$,
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\begin{align*}
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d(x, I_N(x))& = d(x, x_{k_N(x)}) = \min_{n \in C_N(x)}d(x, x_n) \\
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&= \min\bracsn{d(x, x_n)|1 \le n \le N, x_n \in \mathcal{A}(x)}
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\end{align*}
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(1, Borel Measurable): Fix $N \in \natp$, then for each $n \in \natp$,
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\begin{align*}
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\bracs{k_N \le n} &= \bigcup_{j = 1}^n \bracs{x \in X \bigg | j \in C_N(x), d(x, x_j) = \min_{m \in C_N(x)}d(x, x_m)} \\
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&= \bigcup_{j = 1}^n \bracs{j \in C_N} \cap \bracs{x \in X \bigg | d(x, x_j) = \min_{m \in C_N(x)}d(x, x_m)} \\
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&= \bigcup_{j = 1}^n\bigcup_{J \subset [N]} \bracs{j \in C_N, J = C_N} \cap \bracs{x \in X \bigg | d(x, x_j) = \min_{m \in J}d(x, x_m)}
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\end{align*}
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Given that $\mathcal{A}$ is Borel measurable, $\bracs{n \in C_N} = \bracs{x_n \in \mathcal{A}(x)}$ is a Borel set for each $1 \le n \le N$. As a result, $\bracs{J = C_N}$ is Borel for each $J \subset [N]$. Thus $\bracs{j \in C_N, J = C_N}$ is Borel for each $1 \le j \le n$ and $J \subset [N]$.
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On the other hand, for each $1 \le n \le N$, the function $x \mapsto d(x, x_n)$ is continuous and hence Borel measurable. Similarly, for each $J \subset [N]$, the mapping $\real^J \to \real$ with $\alpha \mapsto \min_{j \in J}\alpha_j$ is also Borel measurable.
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The above facts combined show that $\bracs{k_N \le n}$ is a Borel set, and $k_N: X \to [N]$ is a Borel measurable function. Now, let
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\[
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I_N: X \to \bracsn{x_n|1 \le n \le N} \quad x \mapsto x_{k_N(x)}
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\]
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By assumption that $\seq{x_n}$ are distinct, $\bracs{I_N = x_n} = \bracs{k_N = n}$ is a Borel set for each $1 \le n \le N$. Therefore $I_N$ is Borel measurable.
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(1, $\mathcal{A}$-Admissible): Let $N \in \natp$ and $x \in X$, then
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\[
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I_N(x) = x_{k_N(x)} \in \bracs{x_n|n \in C_N(x)} \subset \mathcal{A}(x)
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\]
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(1, Approximation): Let $x \in X$ and $\eps > 0$. Since $x \in \ol{\mathcal{A}(x)^o}$ and $\seq{x_n}$ is dense in $X$, there exists $N_0 \in \natp$ such that $x_{N_0} \in \mathcal{A}(x)$ and $d(x, x_{N_0}) < \eps$. By (3),
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\[
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\limv{N}d(x, I_N(x)) = \limv{N}\min\bracs{d(x, x_n)| 1 \le n \le N, x_n \in \mathcal{A}(x)} = 0
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\]
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\end{proof}
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\begin{remark}
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\label{remark:separable-metric-space-approx-identity}
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In \autoref{lemma:separable-metric-space-approx-identity}, if $X$ is compact and $\mathcal{A} \equiv X$, then $I_N \to I$ \textit{uniformly}.
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\end{remark}
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\begin{corollary}
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\label{corollary:measurable-simple-separable}
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Let $(X, \cm)$ be a measurable space, $Y$ be a separable metric space, and $\mathcal{A}: Y \to 2^Y$ be a Borel measurable \hyperref[admissible approximant function]{definition:admissible-approximant-function}, then for any $f: X \to Y$, the following are equivalent:
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\begin{enumerate}
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\item $f$ is $(\cm, \cb_Y)$-measurable.
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\item For any dense subset $\seq{y_n} \subset Y$ with $y_1 \in \bigcap_{y \in Y}\mathcal{A}(y)$, there exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that
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\begin{enumerate}
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\item[(i)] For each $x \in X$ and $N \in \natp$,
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\[
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f_N(x) \in \mathcal{A}(f(x)) \cap \bracsn{y_n|1 \le n \le N}
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\]
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\item[(ii)] $f_n \to f$ pointwise as $n \to \infty$.
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\end{enumerate}
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\item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that $f_n \to f$ pointwise.
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\end{enumerate}
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\end{corollary}
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\begin{proof}
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(1) $\Rightarrow$ (2): Let $\seq{y_n} \subset Y$ be a dense subset with $y_1 \in \bigcap_{y \in Y}\mathcal{A}(y)$. By \autoref{lemma:separable-metric-space-approx-identity}, there exists $\seq{I_n} \subset Y^Y$ such that:
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\begin{enumerate}
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\item $\seq{I_n}$ is an $\mathcal{A}$-admissible \hyperref[approximation of the identity]{definition:approximation-id-measure}.
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\item For each $N \in \natp$, $I_N$ is Borel measurable with $I_N(Y) \subset \bracsn{y_n|1 \le n \le N}$.
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\end{enumerate}
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For each $n \in \natp$, let $f_n = I_N \circ f_n$, then:
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\begin{enumerate}
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\item[(i)] For each $x \in X$ and $N \in \natp$,
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\[
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f_N(x) = I_N(f(x)) \in \mathcal{A}(f(x)) \cap \bracsn{y_n|1 \le n \le N}
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\]
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\item[(ii)] Since $I_n \to \text{Id}$ pointwise as $n \to \infty$, $f_n \to f$ pointwise as $n \to \infty$.
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\end{enumerate}
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(3) $\Rightarrow$ (1): By \autoref{proposition:metric-measurable-limit}.
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\end{proof}
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\begin{corollary}
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\label{corollary:measurable-simple-separable-norm}
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Let $(X, \cm)$ be a measurable space, $(E, \norm{\cdot}_E)$ be a separable normed vector space, and $f: X \to E$, then the following are equivalent:
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\begin{enumerate}
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\item $f$ is $(\cm, \cb_E)$-measurable.
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\item There exists simple functions $\seq{f_n}$ such that $\abs{f_n} \le \abs{f}$ for all $n \in \natp$, and $f_n \to f$ pointwise.
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\end{enumerate}
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\end{corollary}
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\begin{proof}
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(1) $\Rightarrow$ (2): Let
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\[
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\mathcal{A}: E \to 2^E \quad y \mapsto \begin{cases}
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B_E(0, \norm{y}_E) & y \ne 0 \\
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E & y = 0
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\end{cases}
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\]
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then
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\begin{enumerate}
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\item[(AA1)] For each $y \in E$, $y \in \ol{\mathcal{A}(y)^o}$.
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\item[(AA2)] $0 \in \bigcap_{y \in E}\mathcal{A}(y)$.
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\item[(B)] For any fixed $y_0 \in E \setminus \bracs{0}$,
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\[
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\bracs{y \in E|y_0 \in \mathcal{A}(y)} = \bracs{y \in E|\norm{y_0}_E < \norm{y}_E} \cup \bracs{0} \in \cb_E
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\]
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and $\bracs{y \in E|0 \in \mathcal{A}(y)} = E$.
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\end{enumerate}
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so $\mathcal{A}$ is a Borel measurable \hyperref[admissible approximant function]{definition:admissible-approximant-function}.
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By (2) of \autoref{corollary:measurable-simple-separable}, there exists simple functions $\seq{f_n}$ such that $|f_n| \le |f|$ on $\bracs{f \ne 0}$ for all $n \in \natp$ and $f_n \to f$ pointwise. In which case, $|\one_{\bracs{f \ne 0}}f_n| \le |f|$ globally for all $n \in \natp$ and $\one_{\bracs{f \ne 0}}f_n \to f$ pointwise as $n \to \infty$.
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(2) $\Rightarrow$ (1): By \autoref{proposition:metric-measurable-limit}.
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\end{proof}
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