Baire measures on uncountable product spaces
by Arnold W. Miller
As I was just about finished writing this up (October 1998), I discovered
that everything here is already known. See the last section of
R.J.Gardner and W.F.Pfeffer, Borel Measures in Handbook of Set
Theoretic Topology, North-Holland, 1984, 961-1044.
The results mentioned below were already known to Fremlin (Thm 2), Moran,
Kemperman, and Maharam (Thm 1) at least 20 years ago.
Abstract
Thm 1. We show that assuming the continuum hypothesis there exists a
nontrivial countably additive measure on the Baire subsets of the space
reals^{omega_1} which vanishes on each element of an open cover.
Thm 2. Contrariwise, we show that assuming MA+notCH, that there is no such
measure.
We also includes proofs of the even better known:
Thm 3. (Bockstein) If C is a subset of reals^{omega_1} is a closed
G_\delta set, then C is Baire.
Thm 4. (A.H.Stone) The space \reals^{omega_1} is not normal.
Thm 5. The usual product measure on 2^{omega_1} makes all Borel subsets
measurable (not just the Baire sets).