LowerSemicontinuous 2003-10-15 01:05:31 2007-05-09 19:01:26 Definition lower semicontinuous upper semicontinuous lower semi-continuous upper semi-continuous % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\R}[0]{\mathbb{R}} \newcommand{\C}[0]{\mathbb{C}} \newcommand{\N}[0]{\mathbb{N}} \newcommand{\Z}[0]{\mathbb{Z}} \newcommand{\Q}[0]{\mathbb{Q}} % The below lines should work as the command % \renewcommand{\bibname}{References} % without creating havoc when rendering an entry in % the page-image mode. \makeatletter \@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} \makeatother \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} Suppose $X$ is a topological space, and $f$ is a function from $X$ into the extended real numbers $\mathbb{R}^*$; $f:X\to \mathbb{R}^*$. Then: \begin{enumerate} \item If $f^{-1}((\alpha,\infty])=\{x\in X \mid f(x) >\alpha\}$ is an open set in $X$ for all $\alpha\in \mathbb{R}$, then $f$ is said to be {\bf lower semicontinuous}. \item If $f^{-1}([-\infty,\alpha))=\{x\in X \mid f(x) <\alpha\}$ is an open set in $X$ for all $\alpha\in \mathbb{R}$, then $f$ is said to be {\bf upper semicontinuous}. \end{enumerate} In other words, $f$ is lower semicontinuous, if $f$ is continuous with respect to the topology for $\mathbb{R}^*$ containing $\emptyset$ and open sets $$ U(\alpha) = (\alpha,\infty], \quad \quad \alpha\in \mathbb{R}\cup \{-\infty\}. $$ It is not difficult to see that this is a topology. For example, for a union of sets $U(\alpha_i)$ we have $\cup_i U(\alpha_i)=U(\inf \alpha_i)$. Obviously, this topology is much coarser than the usual topology for the extended numbers. However, the sets $U(\alpha)$ can be seen as neighborhoods of infinity, so in some sense, semicontinuous functions are "continuous at infinity" (see example 3 below). \subsubsection{Examples} \begin{enumerate} \item A function $f\colon X\to \mathbb{R}^*$ is continuous if and only if it is lower and upper semicontinuous. \item Let $f$ be the characteristic function of a set $\Omega\subseteq X$. Then $f$ is lower (upper) semicontinuous if and only if $\Omega$ is open (closed). This also holds for the function that equals $\infty$ in the set and $0\,$ outside. It follows that the characteristic function of $\Q$ is not semicontinuous. \item On $\mathbb{R}$, the function $f(x)=1/x$ for $x\neq 0$ and $f(0)=0$, is not semicontinuous. This example illustrate how semicontinuous "at infinity". \end{enumerate} \subsubsection{Properties} Let $f\colon X\to \mathbb{R}^*$ be a function. \begin{enumerate} \item Restricting $f$ to a subspace preserves semicontinuity. \item Suppose $f$ is upper (lower) semicontinuous, $A$ is a topological space, and $\Psi\colon A\to X$ is a homeomorphism. Then $f\circ\Psi$ is upper (lower) semicontinuous. \item Suppose $f$ is upper (lower) semicontinuous, and $S\colon \mathbb{R}^*\to \mathbb{R}^*$ is a sense preserving homeomorphism. Then $S\circ f$ is upper (lower) semicontinuous. \item $f$ is lower semicontinuous if and only if $-f$ is upper semicontinuous. \end{enumerate} \begin{thebibliography}{9} \bibitem{rudin_real} W. Rudin, \emph{Real and complex analysis}, 3rd ed., McGraw-Hill Inc., 1987. \bibitem{cohn} D.L. Cohn, \emph{Measure Theory}, Birkh\"auser, 1980. \end{thebibliography}