## Inner Measures

A quick one for today.

In analogy with the outer measure induced on the hereditary -ring by the measure on the -ring , we now define the “inner measure” that induces on the same hereditary -ring . We’ve seen that the outer measure is

Accordingly, the inner measure is a set function defined by

In a way, the properties of are “dual” to those of . The easy ones are the same: it’s non-negative, monotone, and .

We could also define in terms of the completed measure. Since , it’s clear

On the other hand, the definition of the completion says that for every there is a with and , and so this is actually an equality.