$\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:

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$\mathrm{\varnothing}\in \mathcal{F}$.

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

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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:

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$E\in \mathcal{F}$.

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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 welldefined, as the intersection of any nonempty 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 $\{\mathrm{\varnothing},E\}$.
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 

Canonical name  sigmaalgebra 
Date of creation  20130322 12:00:28 
Last modified on  20130322 12:00:28 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  16 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 28A60 
Synonym  sigmaalgebra 
Synonym  sigma algebra 
Synonym  $\sigma $ algebra 
Synonym  Borel structure 
Synonym  $\sigma $field 
Synonym  sigmafield 
Synonym  sigma field 
Synonym  $\sigma $ field 
Related topic  Algebra2 
Related topic  BorelSigmaAlgebra 
Related topic  MathcalFMeasurableFunction 
Related topic  RingOfSets 
Defines  generated by 