Measure Theory 1.2 : Sigma Algebras and the Borel Sigma Algebra

In this video, I introduce sigma algebras, generating sigma algebras, the Borel sigma algebra, and much more.

Notes : None yet

[jeffreyhowarth]

I studied p-adic for a month. What interested me about p-adics is people where doing things with p-adics, like solving the zeta function, so me liking your clarity on the white board and the all the abstract material about p-adics online and having looked up the zeta function in p-adic form was looking for a enjoyable math language interpreter like yourself.Myself a math hobbies about 8 years. thanks for you math energy.

[duckymomo]

time for a measure theory joke:

when you realize a set is not in the sigma-algebra: my set is immeasurable and my proof is ruined.

[matouspikous]

I love the short format of these videos.
Btw. Am I the only one who gets an advert every 5 minutes?

[deconfinedQPT]

Just wondering how do you learn?, do you study a lot?, how things make sense for you?. I am just very impressed by your capabilities at this age and it is very intriguing for me, did you get your IQ measured?. I am truly sorry if I am getting too personal I am just very interested in understanding intelligent people. Thank you for the content as well but my background is not enough to comprehend with it. I wish you could do some physics as well.

[professionalprocrastinator]

Good content. I just discovered your channel and was favourably impressed by your level of education, knowledge and intellectual maturity at such a young age. Your vids are also remarkably useful to me cause I want to learn the basics of measure theory without wasting too much time into the rigours and cumbersomeness of a uni lecture.



Around 10:00, you start to prove that "given a certain subset of P(X), there exists a sigma-algebra that contains it" and proceed to construct such a sigma-algebra Sigma. You just assumed that it satisfies (i) but can't this be proved directly? Gamma can be considered to be a union of elements of X, so by construction, taking the union with its complement in X should just yield X. X = (Gamma) U bar(Gamma) is an element of Sigma.

[fvveb]

Your videos are really good and impressive!!!

[njchristoffersen]

Don't forget at 1:15 that it is subadditive, that is to say that it is less than or equal too. Unless of course, you claim that A_i's are disjoint. Great stuff!

[ruthauma]

This is impressive. Am doing my degree but it is so hard to understand this concept but you are only thirteen and you understand. Thank you for this video anyway

[arthurgames]

(Obs.: My english is really bad, so please, forgive me if I make a mistake)

Fematika, Im a 12 years students (in august ill do 13) and I really love mathematics (as you can see, my profile pic is hilbert)! Im learning it since last year and I dont know so much about topology, metric spaces and this advanced things, but I know some calculus, set theory and other no so advanced things, and Im TRYING to learn some real analisys, linear algebra and abstract algebra via internet, and you give me a hope that some day ill actually understand category theory, algebraic topology, and this more complicated things! Keep with your work, Im sure that you are also a inspiration for a lot of students like me!

[nounoumath]

8:11 we don't need X to be part of gama, all what we need is that gama is not empty since the new set is closed under complement.

[Pemmykemmy]

Nice explanation. What book(s) do you like for learning measure theory?

[afzalsoomro]

Very impressive can you suggest book for measure theory for self study

[willnewman]

Can you explain what Sigma is?
You say it is the set of all sequences of countable unions and compliments of elements of Gamma. What is meant by sequences? A sequence is typically a function from N to your set, and would thus not be an element of P(X)

[ΕιρηνηΘεοδωρου-θυ]

Hi friend! I have a question! Could you please give an example of a Borel set that is neither Gδ nor Fσ;

[zafdu]

nice video, my teacher mentionned how set-generated sigma algebras constructed (by definition it is the intersection of every sigma algebras that contains the set) but apparently there is another way of constructing it, by adding enough sets to your set of sets (sic)

[TheWitness]

How old are you dude? These vids are amazing.

[ilka]

My man this is amazing! Could you please teach me stochastic calculus instead of my prof?

[Info_rare]

0:39 : the function assigns all elements of segma into R star? Shouldn’t be R+

[muhammadhaider]

If I need to ask about Measure theory. how can i reach out you>

[marylivesong]

You are amazing man ! I am in my bachelor degree and I am learning from you .Surely you will have a brilliant future .please can I be in contact with you, I will surely need you teacher :)