Added a remark on the Singer representation theorem.

src/measure/radon/c0.tex

@@ -169,5 +169,19 @@
     
 \end{proof}
 
+\begin{remark}
+\label{remark:singer-representation}
+    In the proof of \hyperref[Singer's Representation Theorem]{theorem:singer-representation}, the $E^*$-valued measure is constructed pointwise as
+    \[
+    \nu: \cb_X \to E^* \quad \dpn{y,\nu(A)}{E} = \int_{X \times B} \one_A(x) \cdot \dpn{y, \phi}{E} \mu(dx, d\phi)
+    \]
+
+    It may be more tempting to use the strong formulation directly
+    \[
+    \nu: \cb_X \to E^* \quad \dpn{y,\nu(A)}{E} = \int_{X \times B} \phi \cdot \one_A(x) \mu(dx, d\phi)
+    \]
+
+    However, without additional assumptions on $E^*$, $\phi \cdot \one_A(x)$ may not be strongly measurable, which prevents this direct use of the Bochner integral. Thus the weak formulation is a necessary complication in the proof. 
+\end{remark}