The Measure Extension Theorem: Suppose that \(\cal A_0\) is a subring (that is, \(a,b \in \cal A_0  \to a\vee b \in \cal A_0\) and \(a-b \in \cal A_0\)) of a Boolean algebra \(\cal A\) and \(\mu\) is a measure on \(\cal A_0\) (that is, \(\mu:\cal A \to [0,\infty]\), \(\mu(a\vee b) =\mu(a)+\mu(b)\) for \(a\land b = 0\), and \(\mu(0) = 0\).) then there is a measure on \(\cal A\) that extends \(\mu\).