Measure Theory 12 | Carathéodory's Extension Theorem

[@beback_]

This is not only helping with my Analysis class but also with my German learning because of how you structure your sentences.

[@brightsideofmaths]

Interesting :D I also have a German version of this video if this can even help you more.

[@keyvanfardi419]

Yes. He has a nice accent.

[@qinkoo7682]

Thanks from South Korea and look foward to proof of theorem!

[@nskeip]

Thanks a lot for your work! That is great. You know, in 13:57 seems like [a, c) U [d, b) - seems like it is not quite correct. Because semirings just don't know what unions are. But the idea is correct - [a, b) \ [c, d) is still in the semiring. But for another reason: it is because if A is subset of B, there is a finite disjoint family of sets whose disjoint union is A \ B

[@brightsideofmaths]

Yes, this is indeed what I meant there. Thanks!

[@qiaohuizhou6960]

note: the theorem helps to prove the uniqueness and existence of lebesgue measure

[@duckymomo7935]

Yay this is another good theorem

[@jonathanaraldi6354]

man you're the best, I just needed it today

[@adityamanimishra5053]

nice and simple explanation

[@qiaohuizhou6960]

A is a subset of power set 6:20

[@karimkafi1872]

thank you for your great work

[@swapnanilbakshi8493]

great work sir

[@happyhedgehog6450]

Amazing videos, thank you.

[@duckymomo7935]

You will go over radon nikodym right

[@Fastsina]

Thank you so much!

[@harounsuliman1692]

thank you in advance

[@sunilrampuria7906]

Thanks for the video. I am not convinced regarding why we require the countably infinite union of Aj to be in the semiring. Wouldn't the condition (b) of the measure, vacuously hold true if the countably infinite union doesn't belong to the semi-ring? Thanks

[@brightsideofmaths]

You are welcome! I am not sure if I understand you correctly. I gave the definition of a pre-measure. There, part (b) only makes sense if we look at sequences of sets where the infinite union lies in the semiring.

[@alperencil8653]

Hi, at 16:20 by introducing the union of countable family of sets into a semi-ring aren't we simply assuming that it is a Sigma-algebra? This seems contradictory though, since the Pi-system defined by [a,b) is obviously not a sigma-algebra, but only an algebra (or a semi-ring, as referred in the video). Can you explain that part? Thanks.

[@brightsideofmaths]

We just assume the sigma-additivity for sets where the union in in A again. This property says nothing about the semiring itself.

[@MrWater2]

Haaalo!! What's the difference between the concept of semiring and algebra? For example in Chapter 6. ABSTRACT MEASURE AND INTEGRATION THEORY of Stein and Hakarschy (page 270) they defined the premeasure on an algebra. But looks like the same definition of the semiring given here. Thank you again!

[@metalore]

A Borel Sigma Algebra contains every open, closed and partially open/closed set on R, correct (as a result of the complements and unions of open sets)? I can't find a video where you've shown what exactly is in B(R) (and what is not). For condition b of the Extension Theorem, we just create a sequence of intervals in R that includes all of R? Such as (-n,n) ? When could such a sequence not be found?

[@brightsideofmaths]

I have a video where we construct a non-measurable set.

[@kkkk-oy9qv]

Thank you, you are the best

[@roamlag]

Thanks a lot for this video! (and this entire course) Q: in the most important example, how could you conclude so quickly that \sigma(A) equals the Borel \sigma-algebra?

[@brightsideofmaths]

Glad it was helpful! Indeed, this follows from the definition of the Borel sigma algebra. This is a fact one should remember.

[@novicadakovic6188]

​@@brightsideofmathsIn your video about Borel set you mentioned that this set is sigma algebra generated by open sets. Does this also applies for half-open sets or maybe closed sets in R?

[@brightsideofmaths]

@@novicadakovic6188 Closed sets are complements of open sets and, therefore, also in the sigma algebra :)

[@novicadakovic6188]

@@brightsideofmaths Thank you for your prompt response 😊

[@shibshankardey1262]

Nice Lecture. Plz give some applications of it .

[@brightsideofmaths]

Oh, I have a few applications in the next videos of the series. :)

[@aceofshade]

Is a semiring related to a pi-system?

[@lucynowacki3327]

Excellente. I love your lectures. Can we have Hahn-Banach theorem?

[@sk8aholic123456789]

Is this part of the Measure Theory sequence?

[@brightsideofmaths]

Yes, of course! I put in the Playlist. However, I stopped numbering the videos because you don't have to know all the videos before this one to understand the recent topic :)

[@sk8aholic123456789]

At the risk of stating the obvious, I love your work!

[@aristodemosii.1980]

Did you also upload the 285 videos proving Carathéodory?

[@brightsideofmaths]

If there is demand, yes!

[@MrWater2]

@@brightsideofmaths Please :) !

[@Fastsina]

Monsieur, you said in your video of Product measure and Cavalieri's theorem , that in this video , we've seen that product measure is in general is not unique, did you mean that uniqueness of extension does not apply for product measure?

[@brightsideofmaths]

For uniqueness of the extension you need the sigma-finiteness.

[@Fastsina]

@@brightsideofmaths Thank you!

[@beback_]

He's a Herr not a Monsieur hehe

[@raulfiligranavillalba7605]

Hi, my name's Raúl. I'm trying to prove the Carathéodory's extension theorem and I'm finding some problems with the proof. It's obvious that the demonstration is based on the definition of the outer measure: mu*(A)={sum mu(Sn) | Sn in S for all n, A C U Sn} There's a problem with this definition because, by the definition of a semirring, we cannot be sure that exists a countable family of sets {Sn} which covers X. How do I have to proceed? Any ideas?

[@cartmansuperstar]

Isn´t that the property of sigma-finiteness of the semiring, that was mentioned in the video ?