Extension of measures on two disjoint rings

We present a method of starting with two measures mu(1) and mu(2) defined on disjoint rings R-1 and R-2 respectively, and defining another measure which is an extension of both mu(1) and mu(2). The method is a generalization of a classical method of extending a measure. An outer measure mu* is defined on a hereditary sigma-ring H(R-1, R-2) which is a superset of both R-1 and R-2, and then a complete measure on the class of mu*-measurable sets that belong to H(R-1, R-2) is defined in terms of mu*.