MeasurableSpace 2001-11-11 17:38:01 2006-09-01 00:49:08 Definition measurable set \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} \PMlinkescapeword{states} A \emph{measurable space} is a set $E$ together with a collection $\mathcal{B}$ of subsets of $E$ which is a sigma algebra. The elements of $\mathcal{B}$ are called \emph{measurable sets}. A measurable space is the correct object on which to define a measure; $\mathcal{B}$ will be the collection of sets which actually have a measure. We normally want to ensure that $\mathcal{B}$ contains all the sets we will ever want to use. We usually cannot take $\mathcal{B}$ to be the collection of all subsets of $E$ because the axiom of choice often allows one to construct sets that would lead to a contradiction if we gave them a measure (even zero). For the real numbers, Vitali's theorem states that $\mathcal{B}$ cannot be the collection of all subsets if we hope to have a measure that returns the length of an open interval.