## Measurable Spaces, Measure Spaces, and Measurable Functions

We’ve spent a fair amount of time discussing rings and -rings of sets, and measures as functions on such collections. Now we start considering how these sorts of constructions relate to each other.

A “measurable space” is some set and a choice of a -ring of subsets of . We call the members of the “measurable sets” of the measurable space. This is not to insinuate that is the collection of sets measurable by some outer measure , nor even that we can define a nontrivial measure on in the first place. Normally we just call the measurable space by the same name as the underlying set and omit explicit mention of .

Since it would be sort of silly to have points that can’t be discussed, we add the assumption that every point of is in *some* measurable set. Commonly, it’s the case that itself is measurable — — but we won’t actually require that be a -algebra.

A “measure space” is a measurable space along with a choice of a measure on the -ring . As before, we will usually call a measure space by the same name as the underlying set and omit explicit mention of the -ring and measure. Measure spaces inherit adjectives from their measures; a measure space is called finite, or -finite, or complete if its measure is finite, -finite, or complete, respectively.

Given a measure space , we will routinely use without comment the associated outer measure and inner measure on the hereditary -ring .

As an underlying set equipped with a particular collection of “special” subsets, a measurable space should remind us of a topological space, and like topological spaces they form a category. Remember that our original definition of a continuous function: given topological spaces and , a function is continuous if the preimage of any open set is open — for any we have .

We define a “measurable function” similarly: given measurable spaces and , a function is measurable if the preimage of any measurable set is measurable — for any we have . It’s straightforward to verify that the collection of measurable spaces and measurable functions forms a category. We will set this category in its full generality aside for the moment, as is the usual practice in measure theory, but we will refer to it if appropriate to illuminate a point.

Before I close, though, I’d like to put out a question that I *don’t* know the answer to, and which some friends haven’t really been able to answer when I mused about it in front of them. When we dealt with topology, we were able to recast the basic foundations in terms of nets. That is, a function is continuous if and only if it “preserves limits of convergent nets” — if it sends any convergent net in the domain to another convergent net in the range, and the action of the function commutes with passage to the limit. I like this because the idea of “preserving” some structure (albeit an infinite and often-messy one) feels more natural and algebraic that the idea of inspecting preimages of open sets. And so I put the question out to the audience: what is “preserved” by a measurable function, in the same way that continuous functions preserve limits of convergent nets?