Last time we discussed how to define a measurable subspace of a measurable space in the easy case when is itself a measurable subset of : .
But what if isn’t measurable as a subset of ? To get at this question, we introduce the notion of a “thick” subset. We say that a subset of a measure space is thick if for all measurable . If is itself measurable (as it often is), this condition reduces to asking that . If, further, , then we ask that . As an example, the maximally nonmeasurable set we constructed is thick.
Now I say that if is a thick subset of a measure space , if consists of all intersections of with measurable subsets of , and if is defined by , then is a measure space. This definition of is unambiguous, since if and are two measurable subsets of with , then . The thickness of implies that , and we know that
Since , the second term must be zero, and so . Therefore, , and is indeed unambiguously defined.
Now given a pairwise disjoint sequence of sets in , define to be measurable sets so that . If we define
then we find
and so . Therefore
which shows that is indeed a measure.
WordPress seems to have cleaned up its mess for now, so I’ll try to catch up.
When we’re considering the category of measurable spaces it’s a natural question to ask whether a subset of a measurable space is itself a measurable space in a natural way, and if this constitutes a subobject in the category. Unfortunately, unlike we saw with topological spaces, it’s not always possible to do this with measurable spaces. But let’s see what we can say.
Every subset comes with an inclusion function . If this is a measurable function, then it’s clearly a monomorphism; our question comes down to whether the inclusion is measurable in the first place. And so — as we did with topological spaces — we consider the preimage of a measurable subset . That is, what points satisfy ? Clearly, these are the points in the intersection . And so for to be measurable, we must have be measurable as a subset of .
An easy way for this to happen is for itself to be measurable as a subset of . That is, if , then for any measurable , we have . And so we can define to be the collection of all measurable subsets of that happen to fall within . That is, if and only if and . If is a measure space, with measure , then we can define a measure on by setting . This clearly satisfies the definition of a measure.
Conversely, if is a measure space and , we can make into a measure space ! A subset is in if and only if , and we define for such a subset .
As a variation, if we already have a measurable space we can restrict it to the measurable subspace . If we then define a measure on , we can extend this measure to a measure on by the same definition: , even though this is not the same one as in the previous paragraph.
WordPress seems to have messed with again, and fouled it all up. Dozens of perfectly well-formed expressions are throwing errors, including $ latex \sigma$. Since writing lowercase sigmas is pretty much essential for the current topics, I’m just not going to write until they fix their mess.
And if anyone from WordPress reads this: one of the major I encouraged people to start math, physics, and computer science oriented weblogs on WordPress’ platform is exactly its support for LaTeX. It really annoys me to no end that you keep screwing with it and breaking it in pretty severe ways.