| Version 2 |
Version 1 |
| {\bf Definition} \cite{mukherjea, friedman, folland} |
{\bf Definition} \cite{mukherjea, friedman, folland} |
| Let $X$ be a set, and let $\mathcal{P}(X)$ be the |
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 |
power set of $X$. An \emph{outer measure} on $X$ is a function |
| $\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ satisfying the properties |
$\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ satisfying the properties |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\mu^\ast(\emptyset)=0$. |
\item $\mu^\ast(\emptyset)=0$. |
| \item If $A\subset B$ are subsets in $X$, then $\mu^\ast(A)\le \mu^\ast(B)$. |
\item If $A\subset B$ are subsets in $X$, then $\mu^\ast(A)\le \mu^\ast(B)$. |
| \item If $\{A_i\}$ is a countable collection of subsets of $X$, |
\item If $\{A_i\}$ is a countable collection of subsets of $X$, |
| then |
then |
| $$ \mu^\ast(\bigcup_i A_i) \le \sum_i \mu^\ast (A_i).$$ |
$$ \mu^\ast(\bigcup_i A_i) \le \sum_i \mu^\ast (A_i).$$ |
| \end{enumerate} |
\end{enumerate} |
|
|
| Here, we can make two remarks. First, from (1) and (2), it follows |
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, |
that $\mu^\ast$ is a positive function on $\mathcal{P}(X)$. Second, |
| property (3) also holds for any finite collection of subsets since |
property (3) also holds for any finite collection of subsets since |
| we can always append an infinite sequence of empty sets to |
we can always append an infinite sequence of empty sets to |
| such a collection. |
such a collection. |
|
|
| {\bf Examples} |
{\bf Examples} |
| \begin{itemize} |
\begin{itemize} |
| \item \cite{mukherjea, friedman} |
\item \cite{mukherjea, friedman} |
| On a set $X$, let us define $\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ as |
On a set $X$, let us define $\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ as |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \mu^\ast (E) &=& \left\{ \begin {array}{ll} |
\mu^\ast (E) &=& \left\{ \begin {array}{ll} |
| 1 & \mbox{when}\,\, E\neq \emptyset, \\ |
1 & \mbox{when}\,\, E\neq \emptyset, \\ |
| 0 & \mbox{when}\,\, E=\emptyset. \\ \end{array} \right. |
0 & \mbox{when}\,\, E=\emptyset. \\ \end{array} \right. |
| \end{eqnarray*} |
\end{eqnarray*} |
| Then $\mu^\ast$ is an outer measure. |
Then $\mu^\ast$ is an outer measure. |
| \item \cite{mukherjea} |
\item \cite{mukherjea} |
| On a uncountable set $X$, let us define $\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ as |
On a uncountable set $X$, let us define $\mu^\ast:\mathcal{P}(X)\to [0,\infty]$ as |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \mu^\ast (E) &=& \left\{ \begin {array}{ll} |
\mu^\ast (E) &=& \left\{ \begin {array}{ll} |
| 1 & \mbox{when}\,\, E\, \mbox{is uncountable}, \\ |
1 & \mbox{when}\,\, E\, \mbox{is uncountable}, \\ |
| 0 & \mbox{when}\,\, E\, \mbox{is countable}. \\ \end{array} \right. |
0 & \mbox{when}\,\, E\, \mbox{is countable}. \\ \end{array} \right. |
| \end{eqnarray*} |
\end{eqnarray*} |
| Then $\mu^\ast$ is an outer measure. |
Then $\mu^\ast$ is an outer measure. |
| \end{itemize} |
\end{itemize} |
|
|
| {\bf Theorem} \cite{mukherjea, friedman, folland} |
{\bf Theorem} \cite{mukherjea, friedman, folland} |
| Let $X$ be a set, and let |
Let $X$ be a set, and let |
| $\mathcal{F}$ be a collection of subsets of $X$ such |
$\mathcal{F}$ be a collection of subsets of $X$ such |
| that $\emptyset \in \mathcal{F}$ and $X \in \mathcal{F}$. Further, let |
that $\emptyset \in \mathcal{F}$ and $X \in \mathcal{F}$. Further, let |
| $\rho:\mathcal{F}\to [0,\infty]$ be a mapping such that $\rho(\emptyset)=0$. |
$\rho:\mathcal{F}\to [0,\infty]$ be a mapping such that $\rho(\emptyset)=0$. |
| If $A\subset X$, let |
If $A\subset X$, let |
| $$\mu^\ast(A) = \inf \sum_{i=1}^\infty \rho(F_i),$$ |
$$\mu^\ast(A) = \inf \sum_{i=1}^\infty \rho(F_i),$$ |
| where the infimum is taken over all collections $\{F_i\}_{i=1}^\infty \subset \mathcal{F}$ |
where the infimum is taken over all collections $\{F_i\}_{i=1}^\infty \subset \mathcal{F}$ |
| such that $A\subset \cup_{i=1}^\infty F_i$. |
such that $A\subset \cup_{i=1}^\infty F_i$. |
| Then $\mu^\ast:\mathcal{P}(X)\to[0,\infty]$ is an outer measure. |
Then $\mu^\ast:\mathcal{P}(X)\to[0,\infty]$ is an outer measure. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{mukherjea} |
\bibitem{mukherjea} |
| A. Mukherjea, K. Pothoven, |
A. Mukherjea, K. Pothoven, |
| \emph{Real and Functional analysis}, |
\emph{Real and Functional analysis}, |
| Plenum press, 1978. |
Plenum press, 1978. |
| \bibitem{friedman} |
\bibitem{friedman} |
| A. Friedman, |
A. Friedman, |
| \emph{Foundations of Modern Analysis}, |
\emph{Foundations of Modern Analysis}, |
| Dover publications, 1982. |
Dover publications, 1982. |
| \bibitem{folland} |
\bibitem{folland} |
| G.B. Folland, \emph{Real Analysis: Modern Techniques and Their Applications}, 2nd ed, John Wiley \& Sons, Inc., 1999. |
G.B. Folland, \emph{Real Analysis: Modern Techniques and Their Applications}, 2nd ed, John Wiley \& Sons, Inc., 1999. |
| \end{thebibliography} |
\end{thebibliography} |
