# $\sigma$-algebra

## Introduction

When defining a measure for a set $E$ we usually cannot hope to make every subset of $E$ measurable. Instead we must usually restrict our attention to a specific collection of subsets of $E$, requiring that this collection be closed under operations that we would expect to preserve measurability. A $\sigma$-algebra is such a collection.

## Definition

Given a set $E$, a $\sigma$-algebra in $E$ is a collection $\mathcal{F}$ of subsets of $E$ such that:

• $\varnothing\in\mathcal{F}$.

• Any union of countably many elements of $\mathcal{F}$ is an element of $\mathcal{F}$.

• The complement of any element of $\mathcal{F}$ in $E$ is an element of $\mathcal{F}$.

## Notes

It follows from the definition that any $\sigma$-algebra $\mathcal{F}$ in $E$ also satisfies the properties:

• $E\in\mathcal{F}$.

• Any intersection of countably many elements of $\mathcal{F}$ is an element of $\mathcal{F}$.

Note that a $\sigma$-algebra is a field of sets that is closed under countable unions and countable intersections (rather than just finite unions and finite intersections).

Given any collection $C$ of subsets of $E$, the $\sigma$-algebra $\sigma(C)$ generated by $C$ is defined to be the smallest $\sigma$-algebra in $E$ such that $C\subseteq\sigma(C)$. This is well-defined, as the intersection of any non-empty collection of $\sigma$-algebras in $E$ is also a $\sigma$-algebra in $E$.

## Examples

For any set $E$, the power set $\mathcal{P}(E)$ is a $\sigma$-algebra in $E$, as is the set $\{\varnothing,E\}$.

A more interesting example is the Borel $\sigma$-algebra (http://planetmath.org/BorelSigmaAlgebra) in $\mathbb{R}$, which is the $\sigma$-algebra generated by the open subsets of $\mathbb{R}$, or, equivalently, the $\sigma$-algebra generated by the compact subsets of $\mathbb{R}$.

Title $\sigma$-algebra sigmaalgebra 2013-03-22 12:00:28 2013-03-22 12:00:28 yark (2760) yark (2760) 16 yark (2760) Definition msc 28A60 sigma-algebra sigma algebra $\sigma$ algebra Borel structure $\sigma$-field sigma-field sigma field $\sigma$ field Algebra2 BorelSigmaAlgebra MathcalFMeasurableFunction RingOfSets generated by