[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]\Omega[/math] be the sample space of an experiment. Let [math]E[/math] be an

event with [math]P(E) \gt 0[/math] and define [math]m_E(\omega)[/math] by [math]m_E(\omega) = m(\omega|E)[/math]. Prove that [math]m_E(\omega)[/math] is a distribution function on [math]E[/math], that is, that [math]m_E(\omega) \geq 0[/math] and that [math]\sum_{\omega\in\Omega} m_E(\omega) = 1[/math]. The function [math]m_E[/math] is called the conditional distribution given [math]E[/math].