Let \(A\) be a Boolean ring, \(B\) a subring \(A\) which majorizes \(A\), \(G\) a Dedekind complete lattice group and \(\mu: B\to G\) a (finitely additive) measure. The author proves: (1) An extension \(\nu: A\to G\) of \(\mu\) is a measure iff \(\nu\) is maximal in the collection of all \(G\)-valued supermodular extensions of \(\mu\). (2) If \(\varphi: A\to G\) is a supermodular extension of \(\mu\), then \(\mu\) has a measure extension \(\nu: A\to G\) such that \(\varphi\leq\nu\).