79 lines
2.4 KiB
Markdown
79 lines
2.4 KiB
Markdown
# Math4121 Lecture 25
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## Continue on Measure Theory
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### Borel Measure
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Finite additivity of Jordan content, i.e. for any $\{S_j\}_{j=1}^N$ pairwise disjoint sets and Jordan measurable, then
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$$
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\sum_{j=1}^N c(S_j)=c\left(\bigcup_{j=1}^N S_j\right)
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$$
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This fails for countable unions.
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#### Definition of Borel measurable
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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
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$$
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m\left(\bigcup_{j=1}^\infty S_j\right) = \sum_{j=1}^\infty m(S_j)
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$$
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#### Definition of Borel measure
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Borel measure satisfies the following properties:
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1. $m(I)=\ell(I)$ if $I$ is open, closed, or half-open interval
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2. countable additivity is satisfied
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3. If $R, S$ are Borel measurable and $R\subseteq S$, then $S\setminus R$ is Borel measurable and $m(S\setminus R)=m(S)-m(R)$
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### Borel sets
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#### Definition of sigma-algebra
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A collection of sets $\mathcal{A}$ is called a sigma-algebra if it satisfies the following properties:
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1. $\emptyset \in \mathcal{A}$
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2. If $\{A_j\}_{j=1}^\infty \subset \mathcal{A}$, then $\bigcup_{j=1}^\infty A_j \in \mathcal{A}$
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3. If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$
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#### Definition of Borel sets
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The Borel sets in $\mathbb{R}$ is the smallest sigma-algebra containing all closed intervals.
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#### Proposition
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The Borel sets are Borel measurable.
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(proof in the following lectures)
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<details>
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<summary>Examples for Borel measurable</summary>
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1. Let $S=\{x\in [0,1]: x\in \mathbb{Q}\}$
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$S=\{q_j\}_{j=1}^\infty=\bigcup_{j=1}^\infty \{q_j\}$ (by countability of $\mathbb{Q}$)
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Since $m[q_j,q_j]=0$, $m(S)=0$.
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2. Let $S=SVC(4)$
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Since $c_e(SVC(4))=\frac{1}{2}$ and $c_i(SVC(4))=0$, it is not Jordan measurable.
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$S$ is Borel measurable with $m(S)=\frac{1}{2}$. (use setminus and union to show)
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</details>
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#### Proposition 5.3
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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 $2^{\mathfrak{c}}$.
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Sketch of proof:
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SVC(3) is Jordan measurable, but $|SVC(3)|=\mathfrak{c}$. so $|\mathscr{P}(SVC(3))|=2^\mathfrak{c}$.
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But for any $S\subset \mathscr{P}(SVC(3))$, $c_e(S)\leq c_e(SVC(3))=0$ so $S$ is Jordan measurable.
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However, there are $\mathfrak{c}$ many intervals and $\mathcal{B}$ is generated by countable operations from intervals.
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