# 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:

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)

<details>
<summary>Examples for Borel measurable</summary>

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.

$S$ is Borel measurable with $m(S)=\frac{1}{2}$. (use setminus and union to show)

</details>

#### 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.