[FOM] Category and Measure
James Hirschorn
James.Hirschorn at univie.ac.at
Mon Sep 17 15:36:31 EDT 2007
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
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