outer measureOuterMeasure22003-07-15 12:31:472005-05-17 02:32:02Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}{\bf Definition} \cite{mukherjea, friedman, folland}
Let $X$ be a set, and let $\mathcal{P}(X)$ be the
power set of $X$. An \emph{outer measure} on $X$ is a function
$\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ satisfying the properties
\begin{enumerate}
\item $\mu^\ast(\emptyset)=0$.
\item If $A\subset B$ are subsets in $X$, then $\mu^\ast(A)\le \mu^\ast(B)$.
\item If $\{A_i\}$ is a countable collection of subsets of $X$,
then
$$ \mu^\ast(\bigcup_i A_i) \le \sum_i \mu^\ast (A_i).$$
\end{enumerate}
Here, we can make two remarks. First, from (1) and (2), it follows
that $\mu^\ast$ is a positive function on $\mathcal{P}(X)$. Second,
property (3) also holds for any finite collection of subsets since
we can always append an infinite sequence of empty sets to
such a collection.
\begin{thebibliography}{9}
\bibitem{mukherjea}
A. Mukherjea, K. Pothoven,
\emph{Real and Functional analysis},
Plenum press, 1978.
\bibitem{friedman}
A. Friedman,
\emph{Foundations of Modern Analysis},
Dover publications, 1982.
\bibitem{folland}
G.B. Folland, \emph{Real Analysis: Modern Techniques and Their Applications}, 2nd ed, John Wiley \& Sons, Inc., 1999.
\end{thebibliography}