[FOM] Transfinite Recursion in Functional Analysis and Measure Theory

[Bartosz Wcisło]

This might be well-known, but suppose that you want to construct a real
function with non-measurable graph. First, such a function cannot have
positive measure by Fubini's theorem. So it's enough to make sure that this
function is not measure 0.

You enumerate all open sets U_alpha in the product of reals of measure less
than 1 (there are only continuum many of them) and you want to construct a
function whose graph avoid all these sets. This can be carried out by
transfinite induction: at alpha-th step you make sure that your domain will
contain x_alpha and that the graph will avoid U_alpha.

Best regards,
Bartosz Wcisło

śr., 30 sty 2019 o 04:51 Kenny Easwaran <easwaran at gmail.com> napisał(a):

> 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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