[FOM] Category and Measure

joeshipman@aol.com joeshipman at aol.com
Mon Sep 17 03:32:50 EDT 2007

Two notions of a "small set of reals" are common in descriptive set 

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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