[FOM] Transfinite Recursion in Functional Analysis and Measure Theory

[Joe Shipman]

In my thesis I gave a very general version of such a construction. In an n-dimensional product of Polish spaces (a weaker criterion than Polish also works), you can construct a set whose n! iterated integrals have arbitrary values, if the continuum is a chain of measure-0 sets (that is, it can be expressed as a transfinite union of measure-0 sets of which any non-cofinal subunion has measure 0). This condition follows from CH or Martin’s axiom or a number of other known independent statements, and contradicts the alternative independent “Fubini axioms” under which any iterated integrals that exist must be equal. The n-dimensional Fubini axiom states that there exist uncountable cardinals k1<k2<...<k_n<=c such that there is a nonmeasurable set of cardinality k1 and for i=2...n there is a set of cardinality k_i which is not the union of k_(i-1) measure-0 sets. These follow from the existence of a real valued measurable cardinal, and are consistent with ZFC.

— JS

Sent from my iPhone

> On Jan 30, 2019, at 11:47 AM, WILLIAM TAIT <williamtait at mac.com> wrote:
> 
> I remember that, in a course on measure theory by Kakutani in 1956, he assumed the continuum hypothesis and proved that a certain kind of measure could be extended from an open set U in Euclidean n-space E to the whole space. The construction was by taking an enumeration x_{alpha} (alpha < gamma for some gamma <=omega_1) of E/U and extending the measure to x_{alpha} at the alpha’th stage—-a transfinite recursion. But I don’t remember the details beyond this. I still have notes from the course, but they are unreadable (at least for me) now.
> 
> Bill Tait
> 
> Sent from my iPad
> 
>> On Jan 29, 2019, at 9:49 AM, Kenny Easwaran <easwaran at gmail.com> wrote:
>> 
>> I'm not sure that this is quite measure theory, but I believe that
>> Cantor invented his concepts of the transfinite in order to prove the
>> Cantor-Bendixson theorem. If you iterate the process of removing
>> isolated points from a set, you must terminate in some countable
>> number of steps, and each step involves removing at most countably
>> many points. The remaining set is either perfect or empty. He noticed
>> that in general, you need to iterate more than just finitely many
>> times or omega many times, but can only need to iterate a countable
>> number of times, and developed the theory of ordinals to do this.
>> 
>> https://en.wikipedia.org/wiki/Derived_set_(mathematics)
>> 
>> Kenny Easwaran
>> 
>>> On Tue, Jan 29, 2019 at 12:07 AM Adam Kolany <dr.a.kolany at wp.pl> wrote:
>>> 
>>> Hi,
>>> 
>>> I would appreciate examples of proofs in FA and MT where Transfinite
>>> Recursion was used.
>>> 
>>> Also "sensible" formulations of TR  in ZF set theory  would be welcome.
>>> 
>>> 
>>> Can you help ?
>>> 
>>> 
>>> regards,
>>> 
>>> Adam Kolany
>>> 
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