Chapter 1: Measure Spaces

1.0. Introductory remarks. 1.1. Definitions of algebra, $\sigma$ -algebra. 1.2. Examples. Borel $\sigma$ -algebras, $\mathcal{B}(\mathcal{S})$ , $\mathcal{B}=\mathcal{B}(\mathbb{R})$ . 1.3. Definitions concerning set functions. 1.4. Definition of measure space. 1.5. Definitions concerning measures. 1.6. Lemma. Uniqueness of extension, $\pi$ -systems. 1.7. Theorem. Carathéodory’s extension theorem. 1.8. Lebesgue measure $\mbox{Leb}$ on $((0,1],\mathcal{B}(0,1])$ . 1.9. Lemma. Elementary inequalities. 1.10. Lemma. Monotone-convergence properties of measures. 1.11. Example/Warning.

-algebras

A collection $\Sigma_{0}$ of subsets of $S$ is an algebra if it contains the whole set $S$ and is closed under complements and finite unions.

$\Sigma_{0}$ is an algebra iff $\Sigma_{0}$ contains $S$ and $(\Sigma_{0};+=\Delta,\cdot=\cap)$ is an algebra over the field $(0,1;+,\cdot)$ in the algebraists’ sense.

A collection $\Sigma$ of subsets of $S$ is a $\sigma$ -algebra if it is an algebra closed under countable unions.

Generated -algebras

For a collection of subsets $\mathcal{C}$ , the $\sigma$ -algebra generated by $\mathcal{C}$ is the smallest $\sigma$ -algebra containing $\mathcal{C}$ , which is the intersection of all $\sigma$ -algebras containing $\mathcal{C}$ .

Borel -algebras

For a topological space $S$ , the Borel $\sigma$ -algebra $\mathcal{B}(S)$ on $S$ is the $\sigma$ -algebra generated by the collection of all open subsets of $S$ .

By definition, we set $\mathcal{B}=\mathcal{B}(\mathbb{R})$ . Then, $\mathcal{B}=\sigma(\pi(\mathbb{R}))$ where $\begin{eqnarray*} \pi(\mathbb{R}) & = & \{(-\infty,x]|x\in\mathbb{R}\}. \end{eqnarray*}$

Measure spaces

A measure space is a triple $(S,\Sigma,\mu)$ where $S$ is a set, $\Sigma$ is a $\sigma$ -algebra on $S$ , and $\mu$ is a measure on $(S,\Sigma)$ , i.e. a function $\mu:\Sigma\to[0,\infty]$ such that $\mu$ is countably additive.

The measure space is finite if $\mu(S)<\infty$ .
The measure space is $\sigma$ -finite if $S=\cup_{n\in\mathbb{N}}S_{n}$ such that for $n\in\mathbb{N}$ , $\mu(S_{n})<\infty$ .

Probability measures, triples and spaces

A probability space is a probability triple $(S,\Sigma,\mu)$ , which is a measure space such that $\mu$ is a probability measure, i.e. $\mu(S)=1$ .

-systems

A $\pi$ -system on $S$ is a collection of subsets of $S$ closed under finite intersections.

Suppose that two measures having the same finite total mass agree on a $\pi$ -system, then they agree on the $\sigma$ -algebra generated by the $\pi$ -system.

(Carathéodory’s extension theorem) A countably additive map into $[0,\infty]$ defined on an algebra on $S$ can be extended to a measure on the $\sigma$ -algebra generated by the algebra. Further, if the map is finite, its extension is unique.

Lebesgue measure

We start with finite unions of intervals, and define a function on them as the total length of the intervals, then show this is a countably additive function, and then define Lebesgue measure as its extension to $\mathcal{B}(\mathbb{R})$ .

Subpages

Chapter 1: Measure Spaces

Chapter 1: Measure Spaces

\sigma -algebras

Generated \sigma -algebras

Borel \sigma -algebras

Measure spaces

Probability measures, triples and spaces

\pi -systems

Lebesgue measure

-algebras

Generated -algebras

Borel -algebras

-systems