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.

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