\section{Approximations with Simple Functions}
\label{section:simple-approx}

\begin{definition}[Admissible Approximant Function]
\label{definition:admissible-approximant-function}
    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:
    \begin{enumerate}[label=(AA\arabic*)]
        \item For each $x \in X$, $x \in \overline{\mathcal{A}(x)^o}$. 
        \item $\bigcap_{x \in X}\mathcal{A}(x) \ne \emptyset$. 
    \end{enumerate}

    and $\mathcal{A}$ is \textbf{Borel measurable} if:
    \begin{enumerate}[label=(AA\arabic*), start=2]
        \item[(B)] For any $x_0 \in X$, $\bracs{x \in X|x_0 \in \mathcal{A}(x)} \in \cb_X$.
    \end{enumerate}
\end{definition}

\begin{lemma}
\label{lemma:admissible-approximant-existence}
    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}. 
\end{lemma}


\begin{definition}[Approximation of the Identity]
\label{definition:approximation-id-measure}
    Let $X$ be a topological space, $\mathcal{A}: X \to 2^X$ be an \hyperref[admissible approximant function]{definition:admissible-approximant-function} and $\net{I} \subset X^X$ be a net, then $\net{I}$ is an \textbf{$\mathcal{A}$-admissible approximation of the identity} if:
    \begin{enumerate}[label=(AI\arabic*)]
        \item For each $x \in X$, $I_\alpha(x) \to x$. 
        \item For each $x \in X$ and $\alpha \in A$, $I_\alpha(x) \in \mathcal{A}(x)$. 
    \end{enumerate}

    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 $\alpha \in A$. 
\end{definition}

\begin{lemma}[Existence of Simple Approximations of the Identity]
\label{lemma:separable-metric-space-approx-identity}
    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:
    \begin{enumerate}
        \item $\seq{I_n}$ is an $\mathcal{A}$-admissible \hyperref[approximation of the identity]{definition:approximation-id-measure}. 
        \item For each $N \in \natp$, $I_N$ is Borel measurable with $I_N(X) \subset \bracsn{x_n|1 \le n \le N}$. 
        \item For each $N \in \natp$ and $x \in X$, 
        \[
        d(x, I_N(x)) = \min\bracs{d(x, x_n)| 1 \le n \le N, x_n \in \mathcal{A}(x)}
        \]

        In particular, $I_N(x) \to x$ as $n \to \infty$. 
    \end{enumerate}
\end{lemma}
\begin{proof}
    By removing duplicate elements from the sequence, assume without loss of generality that for each $m, n \in \natp$ with $m \ne n$, $x_m \ne x_n$. 

    Let $N \in \natp$. For each $x \in X$, let
    \[
    C_N(x) = \bracs{1 \le n \le N| x_n \in \mathcal{A}(x)}
    \]

    Since $x_1 \in \bigcap_{y \in X}\mathcal{A}(y)$, $1 \in C_N(x)$ and $C_N(x) \ne \emptyset$. Now, let
    \[
    k_N(x) = \min\bracs{n \in C_N(x) \bigg | d(x, x_n) = \min_{m \in C_N(x)}d(x, x_m)}
    \]

    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. Then, for each $n \in \natp$, 
    \begin{align*}
        \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)} \\
        &= \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)} \\
        &= \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)} 
    \end{align*}

    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 $J \subset [N]$. 

    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. 
    
    
    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
    \[
    I_N: X \to \bracsn{x_n|1 \le n \le N} \quad x \mapsto x_{k_N(x)}
    \]

    then for each $1 \le n \le N$, $\bracs{I_N = x_n} = \bracs{k_N = n}$ is a Borel set. Thus $I_N$ is Borel measurable. 

    In addition, for each $x \in X$, $k_N(x) \in C_N(x)$, so
    \[
    I_N(x) = x_{k_N(x)} \in \bracs{x_n|n \in C_N(x)} \subset \mathcal{A}(x)
    \]

    and $\seq{I_N}$ satisfies (AI2).

    Finally, let $x \in X$, then by definition of $k_N$, 
    \[
        d(x, I_N(x)) = d(x, x_{k_N(x)}) = \min\bracs{d(x, x_n)| 1 \le n \le N, x_n \in \mathcal{A}(x)}
    \]

    For any $\eps > 0$, since $x \in \ol{\mathcal{A}(x)^o}$, there exists $N_0 \in \natp$ such that $x_{N_0} \in \mathcal{A}(x)$ and $d(x, x_{N_0}) < \eps$. In which case, for any $N \ge N_0$, $N_0 \in C_N(x)$ and $d(x, I_N(x)) \le d(x, x_{N_0}) < \eps$. Thus $I_N(x) \to x$ as $N \to \infty$, and $\seq{I_N}$ satisfies (AI1). 
    
    Therefore $\seq{I_N}$ is an approximation of the identity satisfying (1) and (2). 
\end{proof}

\begin{remark}
\label{remark:separable-metric-space-approx-identity}
    In \autoref{lemma:separable-metric-space-approx-identity}, if $X$ is compact and $\mathcal{A} \equiv X$, then $I_N \to I$ \textit{uniformly}. 
\end{remark}

\begin{corollary}
\label{corollary:measurable-simple-separable}
    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:
    \begin{enumerate}
        \item $f$ is $(\cm, \cb_Y)$-measurable.
        \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
        \begin{enumerate}
            \item[(i)] For each $x \in X$ and $N \in \natp$,
            \[
            f_N(x) \in \mathcal{A}(f(x)) \cap \bracsn{y_n|1 \le n \le N}
            \]
            \item[(ii)] $d(f_n, f) \downto 0$ pointwise as $n \to \infty$.
        \end{enumerate}

        \item There exists a sequence $\seq{f_n}$ of $(\cm, \cb_Y)$-measurable simple functions such that $f_n \to f$ pointwise.
    \end{enumerate}
\end{corollary}
\begin{proof}
    (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:
    \begin{enumerate}
        \item $\seq{I_n}$ is an $\mathcal{A}$-admissible \hyperref[approximation of the identity]{definition:approximation-id-measure}. 
        \item For each $N \in \natp$, $I_N$ is Borel measurable with $I_N(Y) \subset \bracsn{y_n|1 \le n \le N}$. 
    \end{enumerate}

    For each $n \in \natp$, let $f_n = I_N \circ f_n$, then:
    \begin{enumerate}
        \item[(i)] For each $x \in X$ and $N \in \natp$,
        \[
        f_N(x) = I_N(f(x)) \in \mathcal{A}(f(x)) \cap \bracsn{y_n|1 \le n \le N}
        \]
        \item[(ii)] Since $I_n \to \text{Id}$ pointwise as $n \to \infty$, $f_n \to f$ pointwise as $n \to \infty$. 
    \end{enumerate}

    (3) $\Rightarrow$ (1): By \autoref{proposition:metric-measurable-limit}.
\end{proof}

\begin{corollary}
\label{corollary:measurable-simple-separable-norm}
    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:
    \begin{enumerate}
        \item $f$ is $(\cm, \cb_E)$-measurable.
        \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.
    \end{enumerate}
\end{corollary}
\begin{proof}
    (1) $\Rightarrow$ (2): Let
    \[
    \mathcal{A}: E \to 2^E \quad y \mapsto \begin{cases}
        B_E(0, \norm{y}_E) & y \ne 0 \\ 
        E & y = 0
    \end{cases}
    \]

    then
    \begin{enumerate}
        \item[(AA1)] For each $y \in E$, $y \in \ol{\mathcal{A}(y)^o}$.
        \item[(AA2)] $0 \in \bigcap_{y \in E}\mathcal{A}(y)$.
        \item[(B)] For any fixed $y_0 \in E \setminus \bracs{0}$,
        \[
        \bracs{y \in E|y_0 \in \mathcal{A}(y)} = \bracs{y \in E|\norm{y_0}_E < \norm{y}_E} \cup \bracs{0} \in \cb_E
        \]

        and $\bracs{y \in E|0 \in \mathcal{A}(y)} = E$. 
    \end{enumerate}

    so $\mathcal{A}$ is a Borel measurable \hyperref[admissible approximant function]{definition:admissible-approximant-function}. 

    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 $n \to \infty$. 

    (2) $\Rightarrow$ (1): By \autoref{proposition:metric-measurable-limit}.
\end{proof}