[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"
>>
[snip]
> 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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