RETRACTION: SEPARABILITY REQUIRED TO DEFINE CONVERGENCE IN MEASURE

src/fa/lp/ui.tex

@@ -48,7 +48,7 @@
 
 \begin{theorem}[Vitali Convergence Theorem]
 \label{theorem:vitali-convergence}
-    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, and $\fF \subset 2^{L^p(X; E)}$ be a filter, then $\fF$ is Cauchy in $L^p(X; E)$ if and only if:
+    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a separable normed vector space over $K \in \RC$, and $\fF \subset 2^{L^p(X; E)}$ be a filter, then $\fF$ is Cauchy in $L^p(X; E)$ if and only if:
     \begin{enumerate}
         \item[(M)] $\fF$ is locally Cauchy in measure. 
         \item[(UI)] For each $\eps > 0$, there exists $M \ge 0$ and $F \in \fF$ such that
@@ -142,7 +142,7 @@
 
 \begin{corollary}[Dominated Convergence Theorem (In Measure)]
 \label{corollary:dct-filter}
-    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g, h \in L^p(X; \real)$ such that:
+    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a separable normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g, h \in L^p(X; \real)$ such that:
     \begin{enumerate}[label=(\alph*)]
         \item[(M)] $\fF \to g$ locally in measure.
         \item[(D)] There exists $F \in \fF$ such that $|f| \le h$ for all $f \in F$. 
@@ -159,7 +159,7 @@
 
 \begin{lemma}[Scheffé]
 \label{lemma:scheffe}
-    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g \in L^p(X; E)$, then $\fF \to g$ in $L^p(X; E)$ if and only if:
+    Let $(X, \cm, \mu)$ be a measure space, $p \in [1, \infty)$, $E$ be a separable normed vector space over $K \in \RC$, $\fF \subset 2^{L^p(X; E)}$ be a filter, and $g \in L^p(X; E)$, then $\fF \to g$ in $L^p(X; E)$ if and only if:
     \begin{enumerate}
         \item[(M)] $\fF \to g$ locally in measure. 
         \item[(N)] $\lim_{f, \fF} \norm{f}_{L^p(X; E)} = \norm{g}_{L^p(X; E)}$. 

src/measure/measurable-maps/in-measure.tex

@@ -3,7 +3,7 @@
 
 \begin{definition}[In Measure]
 \label{definition:in-measure}
-    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$, let
+    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a separable metric space. For each $\eps, \delta > 0$, let
     \[
     U(\delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu\bracs{d(f, g) > \delta} < \eps}
     \]
@@ -33,7 +33,7 @@
 
 \begin{definition}[Ky Fan Metric]
 \label{definition:ky-fan}
-    Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a metric space, and
+    Let $(X, \cm, \mu)$ be a measure space, $(Y, d)$ be a separable metric space, and
     \[
     \alpha: L^0(X; Y)^2 \to [0, \infty) \quad (f, g) \mapsto \inf\bracs{\eps > 0| \mu\bracs{d(f, g) > \eps} \le \eps} \wedge 1
     \]
@@ -76,7 +76,7 @@
 
 \begin{definition}[Locally In Measure]
 \label{definition:locally-in-measure}
-    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a metric space. For each $\eps, \delta > 0$ and $A \in \cm$ with $\mu(A) < \infty$, let
+    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a separable metric space. For each $\eps, \delta > 0$ and $A \in \cm$ with $\mu(A) < \infty$, let
     \[
     U(A, \delta, \eps) = \bracs{(f, g) \in \mathcal{L}^0(X; Y)| \mu(A \cap \bracs{d(f, g) > \delta}) < \eps}
     \]
@@ -106,7 +106,7 @@
 
 \begin{proposition}
 \label{proposition:convergence-in-measure}
-    Let $(X, \cm, \mu)$ be a semifinite measure space, $(Y, d)$ be a metric space, and $\fF$ be a filter of $(\cm, \cb_Y)$-measurable functions, then $\fF$ is Cauchy in measure if and only if:
+    Let $(X, \cm, \mu)$ be a semifinite measure space, $(Y, d)$ be a separable metric space, and $\fF$ be a filter of $(\cm, \cb_Y)$-measurable functions, then $\fF$ is Cauchy in measure if and only if:
     \begin{enumerate}
         \item[(L)] $\fF$ is locally Cauchy in measure. 
         \item[(T)] For each $\eps, \delta > 0$, there exists $F \in \fF$ and $A \in \cm$ with $\mu(A) < \infty$ such that
@@ -135,7 +135,7 @@
 
 \begin{lemma}
 \label{lemma:ae-in-measure}
-    Let $(X, \cm, \mu)$ be a finite measure space, $(Y, d)$ be a metric space, and $\seq{f_n}$ and $f$ be Borel measurable functions from $X$ to $Y$ such that $f_n \to f$ almost everywhere, then $f_n \to f$ in measure.
+    Let $(X, \cm, \mu)$ be a finite measure space, $(Y, d)$ be a separable metric space, and $\seq{f_n}$ and $f$ be Borel measurable functions from $X$ to $Y$ such that $f_n \to f$ almost everywhere, then $f_n \to f$ in measure.
 \end{lemma}
 \begin{proof}
     Let $\eps > 0$, then for almost every $x \in X$, there exists $N \in \natp$ such that $d(f_n(x), f(x)) < \eps$ for all $n \ge N$, so
@@ -151,7 +151,7 @@
 
 \begin{theorem}
 \label{theorem:cauchy-in-measure-limit}
-    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a complete metric space, then:
+    Let $(X, \cm, \mu)$ be a measure space and $(Y, d)$ be a Polish space, then:
     \begin{enumerate}
         \item For any $\seq{f_n} \subset L^0(X; Y)$ that is Cauchy in measure, there exists $f \in L^0(X; Y)$ and a subsequence $\seq{n_k}$ such that $f_{n_k} \to f$ almost everywhere and in measure.
         \item $L^0(X; Y)$ equipped with the uniform structure of convergence in measure is complete.