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

[@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.

[@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

[@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.

[@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.

[@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

[@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 :)

[@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

[@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! :)

[@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.

[@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!

[@jiahao2709]

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

[@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.

[@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.

[@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.

[@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!

[@Cowux]

Best measure theory video on YT!!!!

[@brightsideofmaths]

Glad it was helpful!

[@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

[@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!

[@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.

[@thedan2]

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

[@DimiqBaba]

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

[@brightsideofmaths]

Best idea :)

[@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.

[@evionlast]

Clearly explained, basic examples, very good

[@sallanmega1]

This is actually a great explanation. Greetings from Spain!

[@Thaizir]

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

[@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...

[@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 😀.

[@idontthinkso5966]

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

[@brightsideofmaths]

Thank you very much :)

[@abhaykumar7626]

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

[@brightsideofmaths]

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

[@КонстантинДемьянов-л2п]

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

[@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.

[@watermelon-s6j]

This video helped me a lot, thank you!

[@marekbalcerzak5255]

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

[@brightsideofmaths]

Thanks :)

[@JuanRodriguez-tr6st]

You are one of the best teachers I’ve had

[@JuanRodriguez-tr6st]

Came here for functional analysis, stayed for measure theory

[@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

[@nathanbarnard7896]

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

[@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!

[@ssmhsasd]

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

[@brightsideofmaths]

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

[@sejuprajapati2005]

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

[@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?

[@brittnihall1688]

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

[@arefpourseyedi8087]

The presentation was amazing. Thank you!

[@brightsideofmaths]

Glad you liked it!

[@YitzharVered]

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

[@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

[@IhrffhhgJgfhhbvf]

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

[@brightsideofmaths]

Nice :)

[@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.

[@rafaelb.333]

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

[@hisxmark]

The length is the absolute value of (b - a) = |(b - a)| = |(a - b)|. In other words, the length is a distance, a scalar greater than zero between any distinct points. So the length of (a - b) is equal to the length of (b - a).

[@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!

[@daviddavini847]

This was incredibly helpful, thanks for the knowledge!

[@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.

[@최주희-n2w]

I love this lecture. Thank you :)

[@RAJ6118]

Beautiful insight of the topic

[@lexluthor6975]

Very insightful explanations. Some people are born lucky!

[@wesolyfoton]

Great explanation. Kudos!

[@tree3868]

Could you make more videos about the hysteresis system that described by measure theory

[@brightsideofmaths]

Sounds like a very good idea. Do you have more details there what you want?

[@axelperezmachado5008]

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

[@TDRT23]

THIS VIDEO IS AMAZING!!

[@pointofview6679]

Thanks so much sir.This is amazing and helpful

[@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.

[@_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

[@jasonlim8387]

Thank you, it was a very helpful video!

[@tropicalpajamas9986]

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

[@td_27]

Beni buraya kadar getiren eğitim sistemimize teşekkür ediyorum.

[@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!

[@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.

[@jatayubaxi4553]

Absolutely clear explanation.

[@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❤️

[@syamalchattopadhyay2893]

Excellent video lecture

[@babumaths8066]

Very useful. Thank you sir.

[@PrajwalDSouza]

Thank You. This is really helpful. One of the best introductions out there. I have small doubt, Given, A = {1 , 2, 3} B = {1 , 2} C = {3} D = {1} E = {2, 3} N = {} sigma algebra F = [A, B, C, D, E, N] Is this a sigma algebra on set M = {1, 2, 3}? C U D doesn't belong to F. But, set of all countable unions, A U B U C U D U E U N = A belongs to F. Little confused with the concept of infinite countable unions. It is obvious that A U B U C U D U E U N = A = M and that it must belong to F from (a) of the definition. So, what's the point of the (c)?

[@mushroomsteve]

F is not a sigma-algebra, since as you pointed out, F is not closed under finite, and hence countable, unions. Being closed under countable unions means that the union of at most a countable number of ANY choice of elements of F, is itself an element of F. The union of all of the elements of F, as you listed, is just one possible countable union. The closure property has to hold for ALL possible finite or countable unions. If there exists a finite or countable union of elements of F, that is not an element of F, then F is not a sigma-algebra.

[@dr_rich_r]

And if you insist on a countable union, you can just union with the empty set infinitely many times, or as mentioned in the video, the same set over and over. CUDUDUDUD...

[@xPlosiveFobx]

Thank you this helped so much!

[@varnita4455]

Thank you.

[@pedromazariegos6536]

I love your work man

[@brightsideofmaths]

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

[@K4moo]

Thank you for the great video!! I think (c) is somewhat redundant because (b) says A and A^c (X\A) must both exist, therefore their union is X, and X is in the Sigma Algebra according to (a), the second veens diagram seems wrong in my understanding.

[@brightsideofmaths]

Thank you! What exactly is wrong about the Venn diagram?

[@K4moo]

@@brightsideofmaths I was thinking (c) is union of all elements in sigma algebra, which should be X. I rewatched this part, I think you actually meant the union of elements in one of the element in sigma algebra.

[@abdulghanialmasri5550]

Man, you are a great teacher 👍

[@brightsideofmaths]

I appreciate that! Thanks :)

[@pk_1320]

Loved it, great video!

[@shraddha5588]

Thank you ! its such a good explanation.

[@brightsideofmaths]

Glad it was helpful!

[@ChloeYang-z3g]

Amazing video! A question: for the plot you drew, should it be P(X) instead of X? As X is a set and P(X) is the set of all subsets, so A should be a part of all subsets.

[@brightsideofmaths]

A is a part of X and an element of P(X). So these are the same pictures but with different visualizations.

[@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?

[@monsijbiswal1104]

Just awesome. Loved it!

[@sebon11]

Great video, man!

[@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 :)

[@armatg]

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

[@tsunningwah3471]

I am no math student and once tried to explain to my friend why it is impossible to pick a rational number on the real number line and here is my explanation. Imagine you are to create a number 0.xxxxxxxxx…by determining its decimal places at random, say drawing a number from 0-9. For example if you draw 3 ,2 and 6, then your number will be 0.326. Since there are infinitely many decimal places, the process goes on forever and you will likely be getting an irrational number. To create a rational number say 1/3 or 0.333333 that means you have to keep picking the same number forever which is impossible if you are picking the numbers at random. Is that a correct and good explanation?

[@brightsideofmaths]

Probability = 0 does not mean "impossible". I have a whole video series about that :) tbsom.de/s/pt

[@Woollzable]

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

[@brightsideofmaths]

Yes, exactly :)

[@evaggelosantypas5139]

Really nice video, just a small note though when giving the definition for the σ algebra one needs not include both the empty set and X since σ algebras are closed under taking the complement of a set

[@brightsideofmaths]

There, you are completely right! However, I really like this definition because one sees the smallest Sigma-Algebra immediately in the first part.

[@delberry8777]

I kinda had the reverse "issue" with this why do we need (b)? It seems this simply follows from (a) because if m(X) exists (as by (a), where m() is 'measure') then for any subset A of X for which m(A) exists m(-A) is simply m(X) - m(A). No? I assume there's some pathological examples of sets X where this doesn't hold. I can't think of any. (which doesn't say much). (This is not a criticism by the way, just my way of trying to understand this).

[@evaggelosantypas5139]

@@NewCalculus alright I'm skipping the insults let's cut to the interesting part which is the math of course. You clearly wanted my attention by insulting me so there you have it explain yourself why is measure theory useless

[@rgoswami]

Fantastic work!

[@Raibows1226]

Thank you 🙏🙏🙏🙏 I finally understood

[@brightsideofmaths]

You are very welcome :)

[@psytno]

thank you really I enjoy these topics

[@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!

[@KJ-ii1hq]

Thank you! You save my life😭😭😭

[@brightsideofmaths]

Thanks :D

[@JTan-fq6vy]

Thanks for the great video! Regrading the definition of sigma-algebra, why do we require the set X is measurable? Would it be possible that that a set X is *not* measurable while the its subsets are measurable? For example, we may not know exactly the volume of the universe but the space of the British Museum is somehow measurable. Any explanation or hint is really appreciated!

[@brightsideofmaths]

But X is a subset of X as well :)

[@bilalghermoul3634]

Many thanks for these good and helpful mini-lectures. And I would like you to direct me to a useful textbook on this theory from your point of view.