## 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.

[…] and measurable kernels. These will allow us to move statements we want to show about outer and inner measures to the realm of proper measures, where we can use nice things like […]

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[…] a subset , we write for the image of under the transformation — . I say that the outer and inner Lebesgue measures are both nicely behaved under the transformation […]

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[…] a measure space , we will routinely use without comment the associated outer measure and inner measure on the hereditary -ring […]

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