Added the spectral integral isomorphism.

src/measure/vector/fin.tex

@@ -118,50 +118,6 @@ Despite not covering the full dual space, the bounded Borel functions still form
 \end{proof}
 
 
-The following is an explicit construction of what should be the Arens product on $C(X; \complex)^{**}$ thanks to the above description of $M_R(X; \complex)$ and its dual.
-
-\begin{proposition}
-\label{proposition:measures-dual-algebra}
-    Let $X$ be a compact Hausdorff space, $J: C(X; \complex) \to C(X; \complex)^{**}$ be the natural embedding, and $B$ be the closed unit ball of $C(X; \complex)^{**}$, then
-    \begin{enumerate}
-        \item There exists a unique weak*-continuous involution $C(X; \complex)^{**}$ such that $J(f^*) = J(f)^*$ for all $f \in C(X; \complex)$, given by
-        \[
-            C(X; \complex)^{**} \to C(X; \complex)^{**} \quad \dpn{\mu, \phi^*}{C(X; \complex)^*} = \ol{\dpn{\mu, \phi}{C(X; \complex)^*}}
-        \]
-        \item There exists a unique seperately weak*-continuous bilinear map on $C(X; \complex)^{**}$ such that $J(fg) = J(f)J(g)$ for all $f, g \in C(X; \complex)$.
-        \item $C(X; \complex)^{**}$ equipped with the above involution and product is a commutative unital $C^*$-algebra. 
-    \end{enumerate}
-\end{proposition}
-\begin{proof}
-    It is sufficient to construct the maps in (1) and (2). Let $\seqi{\mu}$ be a maximal mutually singular family of Radon measures on $X$. Using \autoref{theorem:hilbert-measures-dual} and the \hyperref[Riesz Representation Theorem]{theorem:riesz-radon-c0}, identify
-    \[
-    C(X; \complex)^{*} = M_R(X; \complex) = [l^1(I); L^1(\mu_i; \complex)] \quad C(X; \complex)^{**} = [l^\infty(I); L^\infty(\mu_i; \complex)]
-    \]
-
-    Under the above, $C(X; \complex)$ may be identified as the diagonal
-    \[
-    \bracsn{f \in C(X; \complex)^I|f_i = f_j \forall i, j \in I} \subset [l^\infty(I); L^\infty(\mu_i; \complex)]
-    \]
-
-    which is weak*-dense in $[l^\infty(I); L^\infty(\mu_i; \complex)]$ by \hyperref[Goldstine's Theorem]{corollary:weak-dense-unit-ball}. 
-
-    (1): For each $g \in [l^\infty(I); L^\infty(\mu_i; \complex)]$, let $g^* = \ol g$. For any $\mu \in [l^1(I); L^1(\mu_i; \complex)]$, 
-    \[
-    \dpn{\mu, g^*}{[l^1(I); L^1(\mu_i; \complex)]} = \dpn{f, \ol g}{[l^1(I); L^1(\mu_i; \complex)]} = \ol{\dpn{f, g}{[l^1(I); L^1(\mu_i; \complex)]}}
-    \]
-
-    so the conjugation map is weak*-continuous. 
-
-    (2): Let $f, g \in [l^\infty(I); L^\infty(\mu_i; \complex)]$ and $\mu \in [l^1(I); L^1(\mu_i; \complex)]$, 
-    \[
-        \dpn{\mu, fg}{[l^1(I); L^1(\mu_i; \complex)]} = \dpn{f\mu, g}{[l^1(I); L^1(\mu_i; \complex)]} = \dpn{g\mu, f}{[l^1(I); L^1(\mu_i; \complex)]}
-    \]
-
-    so the composition map is separately weak*-continuous. 
-
-    (3): $[l^\infty(I); L^\infty(\mu_i; \complex)]$ is a commutative unital $C^*$-algebra. 
-\end{proof}
-
 \begin{proposition}
 \label{proposition:space-of-measures-extreme-points}
     Let $X$ be an LCH space and $\cm \subset \overline{B_{M_R(X; \complex)}(0, 1)}$ be a compact convex set such that: