# [FOM] Hahn Banach and the Baire Property

D.R. MacIver drm39 at cam.ac.uk
Sat Mar 5 04:58:41 EST 2005

Sorry, managed to find a reference on my own after all.

The existence of a finitely additive probability measure on P(N) which
vanishes on finite sets is sufficient to give a subset of {0, 1}^N which
lacks the baire property. Specifically (identifying {0, 1}^N with P(N) in
the obvious way) the set { a \in {0, 1}^N : u(a) = 0 } lacks the baire
property.

If you want more details I can either reproduce them here, or you can check
out the reference HAF (page 810, Pincus's Pathology'' ).

Thanks,
David MacIver

On Mar 5 2005, D.R. MacIver wrote:

> Does anyone happen to know (and ideally have a cite for) whether it is
> known if the Hahn Banach theorem implies the existence of a subset of R
> without the Baire Property (in ZF say)? I'm presuming it does, as it
> proves the existence of a non-measurable subset of R, but I can't seem to
> find a reference one way or the other.
>
> (Sorry for the rather analysis-like question, but I think it really is
> still FOM. Especially as it's entirely possible that the only way to
> answer it is to construct a certain model of set theory!)
>
> Thanks,
> David R. MacIver