As usual, let be a ring of sets, be the smallest -algebra containing , and be the smallest hereditary -algebra containing . We’ve asked about the relation between a measure on , the outer measure it induces on , and the measure we get by restricting to . But for now, let’s consider what happens when we start with an outer measure on .
Okay, so we’ve got an outer measure on a hereditary -ring — like . We can define the -ring of -measurable sets and restrict to a measure on . And then we can turn around and induce an outer measure on the hereditary -ring .
Now, in general there’s no reason that these two should be related. But we have seen that if came from a measure (as described at the top of this post), then , and the measure induced by is just back again!
When this happens, we say that is a “regular” outer measure. And so we’ve seen that any outer measure induced from a measure on a ring is regular. The converse is true as well: if we have a regular outer measure , then it is induced from the measure on . Induced and regular outer measures are the same.
Doesn’t this start to look a bit like a Galois connection?