# [FOM] Transfinite Recursion in Functional Analysis and Measure Theory

WILLIAM TAIT williamtait at mac.com
Wed Jan 30 11:47:07 EST 2019

```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

> 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,
>>
>>
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```