AtomMeasureTheory 2007-11-26 18:11:55 2007-11-26 18:11:55 Definition % 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 Let $(X, \mathcal{B},\mu)$ be a measure space. A set $A \in \mathcal{B}$ is called an {\bf atom} if $A$ has positive measure and contains no measurable subsets $B \subset A$ such that $0 < \mu (B)< \mu(A)$. An equivalent definition can be: $A$ has positive measure and for every measurable subset $B \subset A$, either $\mu(B)=0$ or $\mu(A-B)=0$.