# [FOM] Concerning Probability Measures

Harvey Friedman friedman at math.ohio-state.edu
Thu Feb 16 22:09:50 EST 2006

```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"
>
> As I recall, even adding omega_1 random reals and taking L(R) will
> work. One has to interpret "omega-1 random reals" correctly: use the
> product measure on the product of omega_1 copies of [0,1] and then force
> with the sets of positive measure. omega_2 reals will work just as well
> and give, in fact, the same class of models.
>
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?

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?).

I want to do some further correspondence with this mysterious Fields
Medalist before I discuss that situation in detail. I'll get back to this.

Harvey Friedman

```