Measure Theory 1 | Sigma Algebras

[@brightsideofmaths]

Please do the quiz to check if you have understood the topic in this video: thebrightsideofmathematics.com/measure_theory/overview/ There is also a dark version of this video! kzbin.info/www/bejne/rouZan57nJyWmbc

[@NilodeRoock]

"Page not found" ?

[@brightsideofmaths]

@@NilodeRoock I've updated the link now :)

[@NilodeRoock]

@@brightsideofmaths Thank you.

[@suhaibalkhaldi]

Thank you very much sir

[@lehoangsonsg7436]

What you are doing is amazing. I hope you can produce more content in English for non-German speakers.

[@NegativeAccelerate]

I honestly feel like learning German to get access to mroe videos

[@Artonox]

THANK YOU GOD FOR BRINGING YOU INTO THIS WORLD. I finally found someone who can actually teach measure theory online! Ive always had this on my mind (my worst subject in mathematics, because i didnt understand my lecturer), and finally, nearly 8 years later, you made this beautiful video series for me to revisit and you explain very well. I did get a first class in the end, but I really was interested in measure theory and ashamed that i wasn't able to do this well. This serves as a second chance for me!

[@NilodeRoock]

Say things like "THANK YOU GOD FOR BRINGING YOU INTO THIS WORLD." to your parents, spouse or siblings, if you have to. Go and support the content provider financially if you want to say thanks. Just my 0,01c.

[@roger9822]

Amazing mini-course series, it helps a lot to get through probability theory. Although your videos are short and illustrative, you never lose mathematical rigidity. Thank you so much!

[@zhaoyuzhu254]

I literately spent 12 mins on KZbin and understand the whole thing, while I spent 2 hours on my professor's recording and still have no idea what he is talking about. :)

[@brightsideofmaths]

Thanks :)

[@scarlettliu885]

I have the same experience as yours.

[@perkelele]

Summary: A measure is a map of the generalized volume of the subsets of X. Power-set: set of all subsets of a set X. if X = {a,b} then P(X)={empty,X, {a}.{b}} Measurable Sets: We don't need to measure all the subsets we can form, only some of them. Can be the whole power-set, but is useful smaller. Useful because generalizing length in a meaningful way doesn't work for all sets, but only some sets. A is a Sigma Algebra: each element is a measurable set a) Empty set, and Full set are elements of A b) If a subset is measurable then so is its complement c) If every individual countable set is part of the sigma algebra, then the union of all these sets is also in the sigma Algebra To speak of an area of A we need for the sets that make it up to be measurable. So if you take all the individual sets (units) that make it up, you will get the whole. The smallest sigma algebra A = {emptyset, X} it validates all three rules. The largest sigma algebra A = P(X) because it contains all the subsets. In the best case scenario we can measure them all. But this is not the case so we are often between these two cases.

[@mercymuenimwangi]

It reached a point I just had to search measure theory for dummies. This is the best tutorial. I immediately subscribed and turned on notifications. Thank you so much

[@cardinalityofaset4992]

Just a minor technical detail: You can slightly generalize the definition of sigma algebra by excluding the empty set from the first condition. Its presence in the sigma algebra immidiately follows from the fact that X must be measurable and that any complement of a measurable set is also measurable. (X^c = X \ X = 0 => 0 is measurable). Awesome list of vidoes, it´s intuitive and entertaining to watch :)

[@johnnyq4260]

I bet when he wrote that he was thinking about topology.

[@nreceda]

Just found your channel. I am taking a course this semester on Stochastic Processes and as far as I can tell, your explanations are much easier to understand so thank you thank you thank you thank you.

[@gulshanamna3796]

You are the one who really can make students as well as teachers to understand measure theory in real meanings

[@brightsideofmaths]

Thank you very much :)

[@PostModernAlchemist]

I have no idea how youre making this subject so approachable for someone who took real analysis and abstract algebra 10+ years ago, but thank you! This is great!

[@brightsideofmaths]

Wow, thank you! :)

[@lebesgue-integral]

This is a nice video! I took measure theory in undergrad and I loved the subject, although it was so abstract. Your videos definetely will help this make make sense to many people!

[@rmbennet]

I came here a year and a half ago I couldn’t understand any of it after the first one or two videos. it’s remarkably more intuitive after abstract algebra and real analysis. It’s actually really interesting.

[@richardgreenhough]

Tried to understand this from a book and didn't. This video enabled me to grasp this easily. Great video!

[@brightsideofmaths]

Glad it was helpful!

[@Heisenberg8307]

The moment I realised this dude is giving a brief explanation on Measure Theory, I subscribed immediately.

[@jiahao2709]

You are the professor of MIT level, you video lectures should be accepted, respected, appreciated and advocated!!!!!!!!

[@gustafa2170]

I'll come back to this video when I'm stronger. Need more training.

[@brightsideofmaths]

Just farm some EXP on my lower level videos ;)

[@danialdunson]

buy more pots

[@afrolichesmain777]

Wow great explanation for an introduction to sigma algebra. It’s my first time looking at this material. Looking forward to the rest of your videos on Measure theory!

[@thedan2]

Amazing video! Amazing series! Please keep it coming! Measure theory has never been easier to understand. Thank you!!

[@Zubair622]

Extremely marvelous explanation really enjoyed it Please also upload lectures of complex analysis

[@brightsideofmaths]

Already done. See here: tbsom.de/s/ca

[@Zubair622]

Thanks sir

[@harshavajpayee7652]

Now after watching this I can say that measure theory is measurable 😅 Thanks for this wonderful video ❤️

[@RenyxGhoul]

Even more succinct and concise than my lectures but I understand it a lot more. Wow. Thank you!

[@julianvillaquira4127]

Your channel is amazing! Thanks for the videos, they are very helpful to me since I will take a measure theory course next semester. New subscriber 😀.

[@DimiqBaba]

Thank you, straight after the lecture I watch your lectures.

[@brightsideofmaths]

Best idea :)

[@shreykabir]

Thanks a lot... I was interested in measure theory and wanted to learn more about it... This video has helped a lot making it easier for a high school to understand...

[@Thaizir]

Thanks for taking the time to produce this content, it brought me back memories when I was studying this course.

[@garrycotton7094]

Great video :D. This reminds me of Group Theory in a way. Empty set, A in Fancy A is like the identity axiom. Complement of A in Fancy A is like the inverse axiom. Union of A_i in Fancy A is like the closure axiom.

[@axelperezmachado5008]

does this mean that a sigma-algebra is a group under union?

[@deept3215]

@@axelperezmachado5008 It isn't because there is no inverse of union in the sigma algebra

[@axelperezmachado5008]

@@deept3215 True! Haven't realised

[@Sciophile]

@@axelperezmachado5008 It's an abelian group with respect to the operation of symmetric difference.

[@abhaykumar7626]

From India a great respect for you . Your videos are amazing

[@brightsideofmaths]

Glad you like them! :) And thanks for your support!

[@sallanmega1]

This is actually a great explanation. Greetings from Spain!

[@marekbalcerzak5255]

I love to watch your videos to get notion about the subject before reading handbook. Great job !

[@brightsideofmaths]

Thanks :)

[@idontthinkso5966]

My goodness, you really do have a video in everything I'm looking for.

[@brightsideofmaths]

Thank you very much :)

[@mongky9903]

I would like to make a example for better understanding for sigma algebra, correct me if I was wrong. Given X = {1, 2, 3} and sigma algebra A = {∅, X, {1}, {2, 3}} Let insert {2} to A to make A = {∅, X, {1}, {2, 3}, {2}} - But complement of {2} which is A \ {2} = {1, 3} ∉ A --> A not sigma algebra Let continue insert {1, 3} to A to make A = {∅, X, {1}, {2, 3}, {2}, {1, 3}} - But now a union between {1} and {2} is {1, 2} ∉ A --> A not sigma algebra Let add {1, 2} to make A = {∅, X, {1}, {2, 3}, {2}, {1, 3}, {1, 2}} - But complement of {1, 2} is {3} ∉ A --> A not sigma algebra Let add {3} into A making A sigma algebra again and also A is now the power set of X.

[@Cowux]

Best measure theory video on YT!!!!

[@brightsideofmaths]

Glad it was helpful!

[@КонстантинДемьянов-л2п]

My man, you kinda sound like Mimir from God of War. Loving it!

[@ssmhsasd]

Very intuitive explanation, thank you. Very helpful for Engineers 👍

[@brightsideofmaths]

Glad it was helpful! :) And thanks for the support!

[@brittnihall1688]

These videos are amazing and incredibly helpful!! Thank you SO much!!

[@Wynell]

Thank you so much here! I have an exam in a few days and you're literally saving me :)

[@brightsideofmaths]

Happy to help! :) And thanks for the support!

[@sejuprajapati2005]

Thank you so much for all videos. Your teaching skill is amazing.

[@davidwright8432]

Thanks; amazingly clear! I hope I'll be able to follow as easily when after a bit more development, you actually start do things with the ideas!

[@evionlast]

Clearly explained, basic examples, very good

[@tropicalpajamas9986]

Very well explained with straightforward and intuitive examples. We neeeeeeeeed more of this exciting course in Mathematics. Keep it up~!!!

[@axelperezmachado5008]

This channel is amazing! So glad i found it! I subscribed of course

[@giorgiozannini5626]

Thanks man you saved me! Studying Bayesian statistics now and couldn't wrap my head around the whole measure stuff. Thank you very much again!

[@nathanbarnard7896]

Just starting measure-therotic probability theory and these are great :)

[@watermelon-s6j]

This video helped me a lot, thank you!

[@JuanRodriguez-tr6st]

You are one of the best teachers I’ve had

[@JuanRodriguez-tr6st]

Came here for functional analysis, stayed for measure theory

[@ramkumarr1725]

Very nicely explained. And truly innovative to link it up to the quiz. I will try to see the other videos but the first chapter was very good.

[@brightsideofmaths]

Thank you! I also want to do quizzes for the other parts if they are helpful.

[@ramkumarr1725]

@@brightsideofmaths They are helpful for retention of material and application, IMHO. Math is not a spectator sport, IMHO.

[@davidshechtman4746]

Answer me this. Cantors diagonal argument requires a square matrix to be certain that every entry is covered. The matrix (list, which ever. I use the word in a general sense) he proposes is based on permutative recombination. So for the universe of {a, b, c} I create a list of permutations abc, bca, cab, etc. Ordered in the manner of 3 columns and 6 rows. Iteration of one additional element, d, to the universe in consideration {a, b, c, d} will now produce a list with 4 columns and a page and a half of rows. The initial Alelph null of basic infinity we are guaranteed by Zermelo is won by Iteration. Clearly construction of a square matrix based on permutative recombination is impossible. How then, pray tell, does it magically occur for Cantor?

[@daviddavini847]

This was incredibly helpful, thanks for the knowledge!

[@aoxinguo4256]

thank you for sharing this amazing video! definitely love it

[@brightsideofmaths]

Thank you for your support :)

[@arefpourseyedi8087]

The presentation was amazing. Thank you!

[@brightsideofmaths]

Glad you liked it!

[@user-ib4bg9kg5s]

The name is so scary, so we need people like you in this world to make them look less intimidating, thanks for the explanation

[@gym5959]

at 08:15 can you explain what do you mean by countable union of infinitely many sets ?

[@brightsideofmaths]

The index set for the union is given by the natural numbers :)

[@_Navani_]

This is such a helpful video! Now i feel like i can pass measure theory

[@boyzrulethawld1]

What course are u taking this for? 🤔

[@cvdvdfhgh4946]

@@boyzrulethawld1 id guess measure theory

[@최주희-n2w]

I love this lecture. Thank you :)

[@lexluthor6975]

Very insightful explanations. Some people are born lucky!

[@RAJ6118]

Beautiful insight of the topic

[@IhrffhhgJgfhhbvf]

I finally understand why a sigma algebra is the way it is. The drawing made it so clear to me

[@brightsideofmaths]

Nice :)

[@Anteater23]

Would you ever consider making a maths series on the subject of topology? Your videos are brilliant!

[@brightsideofmaths]

Thanks! I want to do that, yes :)

[@Anteater23]

The Bright Side Of Mathematics :) topological spaces just seem harder to visualise than metric spaces for me. Metric spaces felt like a very natural concept.

[@brightsideofmaths]

@@Anteater23 Topological spaces are also very natural. Often the concrete distances between points are not import but just the knowing which one is near or far.

[@Woollzable]

Is X here the same as large Omega (sample space) ? Since X is the complement of the empty set. Thx.

[@brightsideofmaths]

Yes, exactly :)

[@gym5959]

at 7:19 in the venn diagram should it not be P(X) instead of X as A belongs to the A(italic) which has elements from the power set of X ?

[@brightsideofmaths]

A is an element of P(X), but a subset of X.

[@gym5959]

@@brightsideofmaths thank you so much , you are amazing ;)

[@YitzharVered]

I went through this crap in introduction to probability and was totally lost. Thank you for explaining.

[@pointofview6679]

Thanks so much sir.This is amazing and helpful

[@jasonlim8387]

Thank you, it was a very helpful video!

[@martinpuente7526]

amazing videos! I'm a econ student and I'm trying to deepen the subject. You said in 10:26 we need two elements of a subset to form a sigma-algebra. What if the subset is the empty set? that would be one element and it satisfies the three conditions

[@brightsideofmaths]

Thank you! I don't understand your question completely. Can you elaborate on that?

[@Hold_it]

​@@brightsideofmathsThe question basically boils down to: can the empty set be a Sigma Algebra? Meaning our Set X is just the empty set itself. In which case the Sigma Algebra would only consist of one element. The empty set.

[@brightsideofmaths]

Answer: Yes, it's possible but uninteresting ;)@@Hold_it

[@Hold_it]

​@@brightsideofmathsNice thank you❤ The formulation that a sigma algebra needs at least two elements also made me unsure, but I get why it isn't really worth noting that this special case exists.

[@martinpuente7526]

​@@Hold_it yes! that was exactly what I meant thanks

[@Bolvarsdad]

What's the difference between stating that the power set of X = {a,b} is P(x) = {{}, X, {a}, {b}}, and P(x) = {{}, {a}, {b}, {a,b}}? Reading Wikipedia, it told me that the latter notation is used, so I guess these are interchangeable?

[@brightsideofmaths]

There is no difference. Both sets P(X) from you are exactly the same.

[@TDRT23]

THIS VIDEO IS AMAZING!!

[@KJ-ii1hq]

Thank you! You save my life😭😭😭

[@brightsideofmaths]

Thanks :D

[@josmithephraim9078]

Wow!! These explanations are really nice. Immediately subscribed 👌

[@josmithephraim9078]

What books can we refer to understand more about Measure theory, distribution functions, chebyshev lemme etc?

[@brightsideofmaths]

There are a lot of books. I really like Schilling's about measures and other stuff :)

[@josmithephraim9078]

@@brightsideofmaths Danke gut !

[@JJ-fb2lp]

Jesus dude. I have not seen any one that can explain this topic better than you, which can mean two things. 1. They don't understand the topic 100% but is trying to teach someone. 2. They don't know how much to dumb it down for people who are just trying to understand this topic.

[@NewCalculus]

You've never understood it because nonsense cannot be understood, only believed.

[@abdulghanialmasri5550]

Man, you are a great teacher 👍

[@brightsideofmaths]

I appreciate that! Thanks :)

[@sebon11]

Great video, man!

[@syamalchattopadhyay2893]

Excellent video lecture

[@armatg]

This is the first video where i understand what a sigma algebra is, thank you!

[@EebstertheGreat]

Part (a) of the definition is somewhat redundant with part (b). If we assume ∅ ∈ 𝒜 and that (A ∈ 𝒜) → (Aᶜ ∈ 𝒜), then ∅ᶜ = X ∈ 𝒜 by definition, so it is not required to include that in part (a).

[@brightsideofmaths]

You are totally right! I often used redundancy is definitions to make them clearer.

[@uorya]

I have no idea what I am doing. I just graduated. But does these sets have something to do with that set paradox letter?

[@ludwig-music]

9:52 Are all sets of a sigma-algebra called "measurable sets", even the ones that are not measurable? "Proof": (a) the power set may include non-measurable sets, and (b) the power set is always a sigma-algebra; hence, (a) and (b) imply that it is possible that some sets of a sigma-algebra are not measurable, even though they are called measurable sets. Edit: I figured it out: If you already have a measure and some subsets that you can measure, then you should also be able to measure all the additional subsets that are required to make the family of subsets a sigma-algebra. So I think that's what the word "measurable" is referring to.

[@pk_1320]

Loved it, great video!

[@wesolyfoton]

Great explanation. Kudos!

[@pedromazariegos6536]

I love your work man

[@brightsideofmaths]

Glad you enjoy it! And thanks for your support :)

[@Raibows1226]

Thank you 🙏🙏🙏🙏 I finally understood

[@brightsideofmaths]

You are very welcome :)

[@mushroomsteve]

Proposition: A sigma-algebra F is closed under finite intersections. Proof: Let F be a sigma algebra on a set X, and let A and B be elements of F. Then (A n B) = ((A^C u B^C)^C). Therefore, (A n B) is an element of F. Corollary: A sigma-algebra that is closed under arbitrary unions is a topology.

[@sheerrmaan]

Because the complements of A and B must belong to the sigma algebra by condition II. And the union of these two complements also belongs by III. And again the complement of it belong to F.

[@mushroomsteve]

@@sheerrmaan Exactly.

[@yujinlee8188]

This is amazing!! Thank you so much 🙏

[@rivology8423]

Love you’re videos

[@monsijbiswal1104]

Just awesome. Loved it!

[@김용재-l1c]

I have a question. At 10:18 do you mean every set in sigma-algebra is measurable set???

[@brightsideofmaths]

By definition: We *call* the elements of a sigma-algebra _measurable_ .

[@김용재-l1c]

@@brightsideofmaths thanks

[@babumaths8066]

Very useful. Thank you sir.

[@rafaelb.333]

So glad I've found your channel. Which book did you use to study this?

[@mokuscsik]

Do the subsets A_i always need to be disjoint, like you drew it, or is it not necessary according to this definition?

[@brightsideofmaths]

We don't assume disjoint sets in this definition :)

[@mokuscsik]

@@brightsideofmaths ok great, thank you 🙌🏼

[@kusalthapa3570]

which board you are using? I am also interested in making tutorial videos but unable to find the board like yours!

[@brightsideofmaths]

I am using a simple Wacom board :)

[@kusalthapa3570]

@@brightsideofmathsi mean the software☺

[@brightsideofmaths]

@@kusalthapa3570 Xournal :)

[@kusalthapa3570]

@@brightsideofmaths thanks❤️

[@jotajota278]

Extremely clear and nice. Thank you so much! New subscriber here.

[@DeadPool-jt1ci]

what kinda of maths are prerequisites for this ? i'd say i'm fairly decent in calc 1,2,3 / differential equations / linear algebra / prob /stats ,and i ve also completed a real analysis course (first 7 chapters of rudin's principles of mathematical analysis)

[@thomasjefferson6225]

is the power set the base set? also my book states the first rule of sigma algebras differently. A) R is a lebesque measurable subset, but B and C are similar. It really emphisies countable addititive as being important. A measurable subset is built of a coutable number of subets with their intersection being the nullset or something like that. In fact that Believe my book says that C says the intersection of these sets should be the null set. Im in love with it lol.

[@brightsideofmaths]

What is your book?

[@thomasjefferson6225]

@@brightsideofmaths capinski and kopp: measure integral and probably. You bring this into effect in a later video.

[@sarthakgupta1165]

What is the reference book for Measure Theory you are following? I would appreciate it if you could suggest some resources for problems (according to your order of topics).

[@brightsideofmaths]

Hard to answer. There are a lot of good books about measure theory. Do you know Bauer?

[@sarthakgupta1165]

@@brightsideofmaths Yes! Are you following Measure and Integration Theory by Heinz Bauer?

[@brightsideofmaths]

No, but I recommend the book :)@@sarthakgupta1165

[@albert.guedes]

I would like to know what this whiteboard app is. Its look very clean and simple.

[@varnita4455]

Thank you.

[@MrOvipare]

I was reading Cybernetics : or control and communication in the animal and the machine by Norbert Wiener yesterday... I was surprised by the fact that I got totally lost when he used concepts of "measures" in the chapter about groups and statistical mechanics. What game is this!? What are those objects? Turns out I didn't knew shit about measure theory, despite my physics and engineering studies + working in R&D. This is exactly where I needed to land!

[@shraddha5588]

Thank you ! its such a good explanation.

[@brightsideofmaths]

Glad it was helpful!

[@Elektrolite111]

Is there a book you'd recommend to match this content?

[@brightsideofmaths]

You can try out Schilling's book!

[@rgoswami]

Fantastic work!