Update Math4121_L25.md

pages/Math4121/Math4121_L25.md

@@ -60,6 +60,18 @@ Since $m[q_j,q_j]=0$, $m(S)=0$.
 
 Since $c_e(SVC(4))=\frac{1}{2}$ and $c_i(SVC(4))=0$, it is not Jordan measurable.
 
-
+$S$ is Borel measurable with $m(S)=\frac{1}{2}$. (use setminus and union to show)
+
+#### Proposition 5.3
+
+Let $\mathcal{B}$ be the Borel sets in $\mathbb{R}$. Then the cardinality of $\mathcal{B}$ is $2^{\aleph_0}=\mathfrak{c}$. But the cardinality of the set of Jordan measurable sets is $2^{\mathfrak{c}}$.
+
+Sketch of proof:
+
+SVC(3) is Jordan measurable, but $|SVC(3)|=\mathfrak{c}$. so $|\mathscr{P}(SVC(3))|=2^\mathfrak{c}$.
+
+But for any $S\subset \mathscr{P}(SVC(3))$, $c_e(S)\leq c_e(SVC(3))=0$ so $S$ is Jordan measurable.
+
+However, there are $\mathfrak{c}$ many intervals and $\mathcal{B}$ is generated by countable operations from intervals.