Minor sharpening of statements.

src/measure/measurable-maps/approx.tex

@@ -34,7 +34,7 @@
 
 \begin{lemma}[Existence of Simple Approximations of the Identity]
 \label{lemma:separable-metric-space-approx-identity}
-    Let $X$ be a separable and metrisable topological 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:
+    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 $I_N(X) \subset \bracsn{x_n|1 \le n \le N}$. 
@@ -49,9 +49,7 @@
     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 $C_N(x) \ne \emptyset$. 
-    
-    Fix a metric $d: Y \times Y \to [0, \infty)$ and let
+    Since $x_1 \in \bigcap_{y \in X}\mathcal{A}(y)$, $1 \in C_N(x)$ and $C_N(x) \ne \emptyset$. Now, let
     \[
     k_N(x) = \min\bracs{n \in C_N(x) \bigg | d(x, x_n) = \min_{m \in C_N(x)}d(x, x_m)}
     \]
@@ -82,7 +80,7 @@
 
     and $\seq{I_N}$ satisfies (AI2).
 
-    Finally, for each $x \in X$ and $\eps > 0$, 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). 
+    Finally, for each $x \in X$ and $M, N \in \natp$ with $M \le N$, $C_M(x) \subset C_N(x)$, so $d(x, I_M(x)) \ge d(x, I_N(x))$, and $\seq{I_n}$ satisfies (3). In particular, 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)-(3). 
 \end{proof}
@@ -91,7 +89,7 @@
 
 \begin{corollary}
 \label{corollary:measurable-simple-separable}
-    Let $(X, \cm)$ be a measurable space, $Y$ be a separable and metrisable topological 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:
+    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
@@ -100,7 +98,7 @@
             \[
             f_N(x) \in \mathcal{A}(f(x)) \cap \bracsn{y_n|1 \le n \le N}
             \]
-            \item[(ii)] $f_n \to f$ pointwise as $n \to \infty$.
+            \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.