A nonmeasurable set

In this video, I show that there exists a non-measurable subset of the real numbers. In other words, that set is so weird that one can not assign the notion of a size to it. Its construction relies on equivalence classes and the axiom of choice. Enjoy!

[albertemcstein]

At first I was confused about the size of the set not being equal to 0, a constant or infinity but then I realized I confused the size, the measure, with the cardinality of a set. Thanks for showing this interesting proof :)

[hassanalihusseini]

Thank you Dr Peyam for this very instructive video, even it was hard to grasp for me (I will watch it again). But one sentence I will remember always:

"...and infinity is not between 1 and 3."

Thank you so much for spending time on this videos!

[shubhamkhansili]

Dr. Preyam, this video of yours filled me with joy. I was struggling to understand this proof and then watched your video, you have cracked down the proof in a very simple way. It feels great to learn from you. I will be thankful to you if you upload lecture series on Measure theory.

[thebeastofgamrz]

Such a beautiful example of the transcendent nature of the axiom of choice... and of quantifying reality itself. It's funny, the axiom of choice is itself a choice.

[plaustrarius]

The set of sets which do not contain themselves as an element! baha

[SimsHacks]

Great video!!!! You're ideal for filling the gaps we don't have time to cover in class

[brandonchan]

Hi Dr Peyam, Just wanted to thank you for posting such a clear explanation of the proof!

[factsheet]

It turns out that you can prove unmeasurable sets using Ultra Filter Theorem, which is strictly weaker than the Axiom of Choice! (you can get Ultra Filters from Choice but not the other way around).
I actually didn't know this, but we could actually live in a world where unmeasurable sets exist, but not every Vector Space has a basis 😁

[yuvalpaz]

If you want weird sets, then in ZF it is consists that there exists sets without linear order!
If A is infinite set such that there are no 2 disjoint infinite sets B, C such that B union C=A then A does not have linear order

[grantsmith]

This is the wildest proof I've seen in a while. I feel like I need to tell everyone about it

[princeardalan]

Thank you Peyam! Do you have a measure theory playlist? I'd like to watch more of your explanations on this topic.

[Debg]

I am currently taking an analysis course in my theoretical physics master's degree and I find this topics really interesting

[NickKravitz]

Nice demonstration. It seems like there should be one one equivalence set with all rational numbers; the remainder contain all irrationals of the form r + q where r is irrational and q is rational. Therefore, just pick r as the representative of each equivalence set. Next you should demonstrate the Cauchy distribution, which contains an undefined mean and variance.

[amitozazad]

After watching until 4:29, I decided not to believe in axiom of choice.

[Aviationlover-belugaxl]

This video is amazing 😃! Just one question: does this sort of mean that the size is infinitesimal but not zero?

[jacks.]

"There exists a set IN NATURE" lol

[dgrandlapinblanc]

Trop dur pour moi actuellement mais merci quand même...

[tajpa]

Please, is the union of classes ( you put a greek letter) that divide the set [0, 1] equal the set [0, 1] itself?

[pj]

Is an "infinitesimal size" acceptable? Just a size which makes the limit as N goes to infinity of N*C be between 1 and 3?

[xy]

Are there non-mesurable subsets of ℝ without the axiom of choice?