SingularMeasures 2003-02-09 10:36:10 2003-02-10 17:49:53 Definition singular % 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 Two measures $\mu$ and $\nu$ in a measurable space $(\Omega,\mathcal{A})$ are called \emph{singular} if there exist two disjoint sets $A$ and $B$ in $\mathcal{A}$ such that $A\cup B =\Omega$ and $\mu(B)=\nu(A) = 0$. This is denoted by $\mu\perp\nu$.