[FOM] Category and Measure
John McCarthy
jmc at cs.Stanford.EDU
Mon Sep 17 13:30:02 EDT 2007
I was once able to prove that the set of values of a parameter for
giving "untypical" behavior of a transformation depending on the
parameter was meager, i.e. of the first category. I couldn't prove it
was of measure zero. [The transformations were of a circle into
itself, and the "typical" behavior is that for each rational number,
there is an interval of positive length of values of the parameter,
for which the rotation number of the transformation is that rational.]
> Date: Mon, 17 Sep 2007 03:32:50 -0400
> From: joeshipman at aol.com
> 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.)
>
> -- JS
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