Regular Outer Measures
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?