[FOM] Concerning Probability Measures

Robert M. Solovay solovay at Math.Berkeley.EDU
Sat Feb 18 18:30:24 EST 2006

Brief technical comments interspersed below.

On Thu, 16 Feb 2006, Harvey Friedman wrote:

> On 2/16/06 6:35 PM, "Robert M. Solovay" <solovay at math.berkeley.edu> wrote:
> Friedman wrote:
>>> ZFDC + "there is a countably additive probability measure on all subsets of
>>> [0,1]"
>>> and
>>> ZF
>>> are equiconsistent (in fact, mutually interpretable).
>> Yes,this is correct [and my theorem]. It was the byproduct of my
>> first attempt to prove the consistency of "All sets Lebesgue measurable"
>> with DC. I think Sacks published a proof of this under the title
>> "Measure-theoretic uniformity"

> I am sure that you noticed that it is a translation invariant extension of
> Lebesgue measure in these models. Furthermore, I would assume that any
> Lebesgue measure preserving automorphism of R remains measure preserving?

 	Yes to both your questions. I also proved, to my dismay, that this 
extension was not just Lebesgue measure. {Nowadays this would follow from 
results of Shelah saying the inaccessible in my "all sets Lebesgue 
measurable" proof is needed.

> > I interpret your results along these lines as further indication that
> whatever intuition people may have had for being able to measure all sets of
> reals, it was tied up with ideas about sets of reals that are incompatible
> with a well ordering of the reals. Once we have a well ordering of the
> reals, even translation invariance becomes impossible, and the statement
> takes on a very different character. By the other work of yours, it then
> jumps to the level of a measurable cardinal. (I take it that just a well
> ordering of the reals will allow for the reversal?).

 	Yes, if the domain is well-ordered, one cando the obvious 
inner-model construction. One might wind up with just a real-valued 
measurable cardinal, but then another inner-model construction [found in 
my paper on real-valued meaaurable cardinals] gets a model of ZFC with a 
measurable cardinal.

 	In the model under discussion [where there is a translation 
invariant extension of Lebesgue measure] there is a two -valued 
measure on the quotient of the reals mod the rationals. So once again this 
has no real consistency strength if the domain is not wel-orderable.

 	--Bob Solovay

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