[FOM] Category and Measure
Francois G. Dorais
dorais at math.cornell.edu
Sat Sep 22 18:46:02 EDT 2007
joeshipman at aol.com wrote:
> Two notions of a "small set of reals" are common in descriptive set
> theory:
>
> A set is null (or "measure 0") if it can be covered by a sequence of
> intervals of arbitrarily small total length
> A set is meager (or "1st category") if it is a union of countably many
> nowhere-dense sets
>
> Unfortunately for intuition, both of these cannot be thought of as
> "small" because the reals can be expressed as a union of a null set and
> a meager set!
>
> I understand why it make sense to think of null sets as "small", and
> know of many applications of this notion. I also understand that the
> notion of meager set makes sense in arbitrary topological spaces, while
> the notion of null set requires a measure space.
>
> But, can someone explain what's so useful about meager sets when
> working with a measure space like the real numbers?
>
> In other words, what kinds of results of (ordinary real) analysis can
> be proven with arguments about category but not with arguments about
> measure? (The more elementary the statement of the *result*, the better
> -- the *proofs* don't have to be elementary.)
This does not really answer your question, but note that many measure
theoretic results can be recast as category theoretic results on the
Stone space of the Boolean algebra Borel/Null. The translation is
somewhat artificial, but those familiar enough with forcing will be able
to translate a proof of a suitable measure theoretic statement into a
random forcing argument and then as a category theoretic statement over
the Stone space of Borel/Null.
>From the set-theoretic point of view, it is clear that category is the
more powerful tool, provided one allows all relevant base spaces, since
every forcing argument is also a category theoretic argument. However,
this is rather unfair and it is probably more appropriate to restrict to
Polish spaces. In that case, category corresponds to Cohen forcing and
measure corresponds to random forcing. These forcings have been very
well studied and it would be hard to make a case for either one.
Generalizing the ther way, one could accept as a "measure theoretic" any
forcing argument with a cBa that has a positive probability measure on
it. Such forcings are rather nice (e.g., they are all ccc) and far from
trivial. As far as I know, this is one of the least studied notion of
niceness for forcing. Although I can think of some technical problems
with this class of forcings, it is not clear to me why this is the state
of affairs. (It could be that this perception is only a reflection of
my ignorance.)
--
François G. Dorais
Department of Mathematics
Cornell University
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