$\sigma$-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 $ℱ$ of subsets of $E$ such that:

• $\mathrm{\varnothing }\in ℱ$.

• Any union of countably many elements of $ℱ$ is an element of $ℱ$.

• The complement of any element of $ℱ$ in $E$ is an element of $ℱ$.

Notes

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

• $E\in ℱ$.

• Any intersection of countably many elements of $ℱ$ is an element of $ℱ$.

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

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

Title	$\sigma$-algebra
Canonical name	sigmaalgebra
Date of creation	2013-03-22 12:00:28
Last modified on	2013-03-22 12:00:28
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	16
Author	yark (2760)
Entry type	Definition
Classification	msc 28A60
Synonym	sigma-algebra
Synonym	sigma algebra
Synonym	$\sigma$ algebra
Synonym	Borel structure
Synonym	$\sigma$-field
Synonym	sigma-field
Synonym	sigma field
Synonym	$\sigma$ field
Related topic	Algebra2
Related topic	BorelSigmaAlgebra
Related topic	MathcalFMeasurableFunction
Related topic	RingOfSets
Defines	generated by