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-# Lecture 25
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+# Math4121 Lecture 25
+
+## Continue on Measure Theory
+
+### Borel Mesure
+
+Finite additivity of Jordan content, i.e. for any $\{S_j\}_{j=1}^N$ pairwise disjoint sets and Jordan measurable, then
+
+$$
+\sum_{j=1}^N c(S_j)=c\left(\bigcup_{j=1}^N S_j\right)
+$$
+
+This fails for countable unions.
+
+#### Definition of Borel measurable
+
+Borel introduced a new measure, called _Borel measure_, was net only finitely addition, but also _countably additive_, meaning $\{S_j\}_{j=1}^\infty$ pairwise disjoint and Borel measurable, then
+
+$$
+m\left(\bigcup_{j=1}^\infty S_j\right) = \sum_{j=1}^\infty m(S_j)
+$$
+
+#### Definition of Borel measure
+
+Borel measure satisfies the following properties:
+
+1. $m(I)=\ell(I)$ if $I$ is open, closed, or half-open interval
+2. countable additivity is satisfied
+3. If $R, S$ are Borel measurable and $R\subseteq S$, then $S\setminus R$ is Borel measurable and $m(S\setminus R)=m(S)-m(R)$
+
+### Borel sets
+
+#### Definition of sigma-algebra
+
+A collection of sets $\mathcal{A}$ is called a sigma-algebra if it satisfies the following properties:
+
+1. $\emptyset \in \mathcal{A}$
+2. If $\{A_j\}_{j=1}^\infty \subset \mathcal{A}$, then $\bigcup_{j=1}^\infty A_j \in \mathcal{A}$
+3. If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$
+
+#### Definition of Borel sets
+
+The Borel sets in $\mathbb{R}$ is the smallest sigma-algebra containing all closed intervals.
+
+#### Proposition
+
+The Borel sets are Borel measurable.
+
+(proof in the following lectures)
+
+Examples:
+
+1. Let $S=\{x\in [0,1]: x\in \mathbb{Q}\}$
+
+$S=\{q_j\}_{j=1}^\infty=\bigcup_{j=1}^\infty \{q_j\}$ (by countability of $\mathbb{Q}$)
+
+Since $m[q_j,q_j]=0$, $m(S)=0$.
+
+2. Let $S=SVC(4)$
+
+Since $c_e(SVC(4))=\frac{1}{2}$ and $c_i(SVC(4))=0$, it is not Jordan measurable.
+
+
+
+