Borel Sigma Field

Definition

Let $C=\{[a,b]:a,b\in \mathbb{R}\}$ be the collection of all closed intervals or all open intervals. By the ^prop-1-1, we know $\sigma (C)$ exists, and we call it the Borel $\sigma$-field. Any set in the Borel $\sigma$-field is called a Borel set. For a topological space $X$, the Borel $\sigma$-algebra of it is the collection of all Borel set on it. The Borel algebra is the smallest $\sigma$-algebra containing all open sets on $X$.

Borel Measure

Any Measure $\mu$ defined on the Borel algebra is a Borel measure. Typically the choice of Borel measure that assigns $\mu ((a,b])=b-a$ for every half-open interval $(a,b]$ is sometimes called “the” Borel measure on ${\mathbb{R}}^n$¹. This is also called the Lebesgue measure or the uniform measure. Note that the Lebesgue measure also refers to the Complete Measure on $\sigma (\mathcal{B}\cup \mathcal{N})$, where $\mathcal{N}$ is the collection of all null (zero-measure) sets.

Construction of Lebesgue Measure

To formally define the Lebesgue measure, we can use Caratheodory’s Extension. We first define a field (closed under finite union)

$${\mathcal{F}}_0=\left\{{\cup }_{i=1}^n(a_i,b_i]:n\in \mathbb{N},a_i,b_i\in \mathbb{R}\right\}.$$

It can be shown that

• ${\mathcal{F}}_0$ is a field but not a Sigma Field.
• $\sigma ({\mathcal{F}}_0)=\sigma (\mathcal{C})=\mathcal{B}$.

Define ${\mathbb{P}}_0$ as

$${\mathbb{P}}_0({\cup }_{i=1}^n(a_i,b_i])=\sum _{i=1}^n(b_i-a_i).$$

It can be proved (nontrivial) that ${\mathbb{P}}_0$ is countably additive. Then the Lebesgue measure is defined as the Caratheodory’s Extension of ${\mathbb{P}}_0$.

Uniform Distribution on $\{0,1{\}}^{\infty }$

Using the binary representation of real numbers, the Lebesgue measure is equivalent to the uniform distribution on $\{0,1{\}}^{\infty }$, with the Sigma Field $\sigma ({\mathcal{F}}_0)$, where

$${\mathcal{F}}_0=\bigcup _{n=1}^{\infty }{\mathcal{F}}_n=\bigcup _{n=1}^{\infty }\left\{B\times {\left\{0,1\right\}}^{\infty }:B\subset {\left\{0,1\right\}}^n\right\}.$$

That is ${\mathcal{F}}_n$ is the collection of all events whose occurrence can be decided by looking at the results of the first $n$ digits. ${\mathcal{F}}_n$ is a $\sigma$-field while ${\mathcal{F}}_0$ is not. The uniform probability measure is consistent with ${\mathbb{P}}_0$ defined on ${\mathcal{F}}_0$:

$${\mathbb{P}}_0(A)=|B|/2^n.$$

Related Problems

• [[6-7700-hw1#exercise-3-borel—sigma-field|Exercise 3 (Borel $sigma$-field)]]
• Exercise 3

Footnotes

1. When $n>1$, $(a,b]=(a_1,b_1]\times \dots (a_n,b_n]$, $b-a=\prod (b_i-a_i)$. ↩