Extension of Measures

Abstract

We have seen that a measure induces an outer measure and that an outer measure induces a measure, both in a certain natural way. If we start with a measure µ, form the induced outer measure µ*, and then form the measure µ̄ induced by µ*, what is the relation between µ and µ̄? The main purpose of the present section is to answer this question. Throughout this section we shall assume that

µ is a measure on a ring R, µ* is the induced outer measure on H(R), and µ̄ is the measure induced by µ* on the σ-ring S̄ of all µ*-measurable sets.