Measurable Subspaces I
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.