[FOM] Category and Measure

[James Hirschorn]

On Monday 17 September 2007 03:32, joeshipman at aol.com wrote:
> 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.)

I'm not sure exactly what counts as a "result of analysis", since you are 
clearly not interested in results specifically about category, e.g. the Baire 
Category Theorem. But perhaps the following example is relevant:

Let a_n and b_n be sequences of real numbers, indexed by the natural numbers. 
Call them *similar* if one can be obtained from the other by translation and 
dilation, i.e. 

     a_n = r X b_n + s for all n, for some fixed reals r and s.

Consider the following statement: "Given a bounded sequence a_n of reals, 
every Borel set of reals that is not 'small' contains a sequence similar to 
a_n".

This statement with 'small' interpreted as meager is a theorem of Erdos. The 
last I heard (several years ago) it is an open problem for 'small' = Lebesgue 
measure zero.

James Hirschorn