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11
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'$\sigma$-algebra'
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| Title of object: |
$\sigma$-algebra |
| Canonical Name: |
SigmaAlgebra |
| Type: |
Definition |
| Created on: |
2001-11-17 10:49:27 |
| Modified on: |
2007-07-02 10:34:02 |
| Classification: |
msc:28A60 |
| Defines: |
generated by |
| Synonyms: |
$\sigma$-algebra=sigma-algebra
$\sigma$-algebra=sigma algebra
$\sigma$-algebra=$\sigma$ algebra
$\sigma$-algebra=Borel structure
$\sigma$-algebra=$\sigma$-field
$\sigma$-algebra=sigma-field
$\sigma$-algebra=sigma field
$\sigma$-algebra=$\sigma$ field |
Revision comment (for changes between this and next version):
| mention that it's a field of sets |
Preamble:
\usepackage{amssymb}
\def\emptyset{\varnothing}
\def\F{\mathcal{F}}
\def\R{\mathbb{R}}
\def\powerset#1{\mathcal{P}(#1)}
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Content:
\PMlinkescapeword{preserve}
\section*{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.
\section*{Definition}
Given a set $E$, a \emph{$\sigma$-algebra} in $E$
is a collection $\F$ of subsets of $E$ such that:
\begin{itemize}
\item $\emptyset\in\F$.
\item Any union of countably many elements of $\F$
is an element of $\F$.
\item The complement of any element of $\F$ in $E$
is an element of $\F$.
\end{itemize}
\section*{Notes}
It follows from the definition that any $\sigma$-algebra $\F$ in $E$
also satisfies the properties:
\begin{itemize}
\item $E\in\F$.
\item Any intersection of countably many elements of $\F$
is an element of $\F$.
\end{itemize}
Given any collection $C$ of subsets of $E$,
the $\sigma$-algebra $\sigma(C)$ \emph{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 $E$.
\section*{Examples}
For any set $E$,
the power set $\powerset{E}$ is a $\sigma$-algebra in $E$,
as is the set $\{\emptyset,E\}$.
A more interesting example is the
\PMlinkname{Borel $\sigma$-algebra}{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 $\R$. |
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