## 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.

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