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Added first darft of gluing lemma for measurable functions.

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Bokuan Li
2026-06-28 19:50:05 -04:00
parent 3a4f4b46e8
commit 4226adf856
2 changed files with 34 additions and 5 deletions
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6
src/measure/measurable-maps/approx.tex
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@@ -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 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:
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:
\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 $I_N(X) \subset \bracsn{x_n|1 \le n \le N}$.
@@ -50,7 +50,7 @@
Since $x_1 \in \bigcap_{y \in X}\mathcal{A}(y)$, $1 \in C_N(x)$ and $C_N(x) \ne \emptyset$.
From here, let
Fix a metric $d: Y \times Y \to [0, \infty)$ and let
\[
k_N(x) = \min\bracs{n \in C_N(x) \bigg | d(x, x_n) = \min_{m \in C_N(x)}d(x, x_m)}
\]
@@ -90,7 +90,7 @@
\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:
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:
\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