Math4121 Lecture 25

Continue on Measure Theory

Borel Measure

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:

$m(I)=\ell(I)$ if $I$ is open, closed, or half-open interval
countable additivity is satisfied
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:

$\emptyset \in \mathcal{A}$
If $\{A_j\}_{j=1}^\infty \subset \mathcal{A}$ , then $\bigcup_{j=1}^\infty A_j \in \mathcal{A}$
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 for Borel measurable

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 $m[q_j,q_j]=0$ , $m(S)=0$ .

Let $S=SVC(4)$

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.