$\sigma$-algebra SigmaAlgebra 2001-11-17 10:49:27 2007-07-25 11:19:23 Definition generated by \usepackage{amssymb} \def\emptyset{\varnothing} \def\F{\mathcal{F}} \def\R{\mathbb{R}} \def\powerset#1{\mathcal{P}(#1)} \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} 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 $\sigma(C)$ \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 $\R$.