Measure Theory 2 | Borel Sigma Algebras

[@brightsideofmaths]

Please do the quiz to check if you have understood the topic in this video: tbsom.de/s/mt At 2:03 you wanted to say "intersection" (as written) and not "union".

[@user-ms3ox7zi5r]

I literally cried trying to understand what the smallest sigma-algebra means... and then your videos showed up on my recommended... I can't be thankful enough !!

[@noahgilbertson7530]

this is why I came here 😂

[@itsRAWRtime007]

Just discovered your channel. I am not even a mathematician nor a student, I am just interested in math as a hobby and this channel gives the overview of math concepts that I needed. Keep it up.

[@NegativeAccelerate]

You are my saviour. 3blue1brown was super helpful in first year. But I need your channel for more complex stuff and I am forever grateful. I hope when I reach my masters you'll still be creating more and more advanced math videos

[@brightsideofmaths]

Happy to help! :) I will create more and more videos in future!

[@Foo321]

3b1b is good for showing animations, introducing interesting concepts and showing motivation behind them. However personally after watching his videos I usually understand less than before due to lack of rigor :p They serve different purposes - this channel actually learns Math.

[@harrypadarri6349]

@@Foo321Yeah, 3b1b is the one who comes and asks: Have you ever tried this here? It will be a great experience! And bright side of maths comes and offers you the strong stuff. When it comes to mathematics, I fully condone it!

[@parvillner5721]

I have been reading about sigma algebra for a few days now, and I have constantly been wondering why no one explain the concepts with simple examples, where each step is explained, such as how you find the the sigma algebra generated from some set. And why they cannot give simple explanations of terms such as "topology". You are the first to do it in a pedagogic fashion! Brilliant video!

[@PunmasterSTP]

Yeah, I'm on the exact same page. Those examples during the video totally saved me!

[@OkThisllbeMyName]

jeesus this channel saved me. couldn't believe nobody told me what a borel measurable set actually is when it's such a simple thing

[@ngantrantrieuthanh1744]

I'm gonna have measure theory and integration in the next semester, and your videos are just really helpful for me to get a headstart! Thank you so much.

[@PunmasterSTP]

How did those classes go?

[@구원-p7l]

This video explains why we need internet despite of all the mess resides in it. I was having trouble understanding how topology has something to do with measure theory but you mentioned it very explicitly. thanks alot!

[@PunmasterSTP]

I completely agree, and I don't think Logan Paul was gonna bust out a video on measure theory anytime soon. 😀

[@luvsharma4655]

Welcome To measure theory! I wish my professor was this happy while teaching!! :D

[@UcritGardening]

Me too😭

[@PunmasterSTP]

How did your class go?

[@danh5995]

The best youtube channel. This is how the internet meant to be, sharing knowledge and enlightening others😀

[@ryanhutchins2634]

100%

[@PunmasterSTP]

I completely agree!

[@tyw6864]

I'm using your channel to study my phd exam ..... thank you so much !!

[@JorgeMartinez-pc7so]

Lovely explanation, I found it veeeeery helpful; I had spent days trying to make sense out of this and it turned out to be this simple!

[@teacherabdalsalamalkhateeb3005]

This section is fun and full of ideas about measurable sets

[@thedan2]

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

[@Capitalust]

This is definitely the go to youtube playlist for measure theory

[@aarohgokhale8832]

2:55 reminds me a lot of taking closures in topological spaces. It follows the same idea of intersecting all closed sets containing the set we want the closure of.

[@utkarshtrehan9128]

I am master's student and my professor tell me that you have already seen Lebesgue Integration in High School 🔥

[@PunmasterSTP]

How are your studies going?

[@bradleymorris161]

I wish I had these videos when I was studying Measure theory 10 years ago. Fantastic video.

[@brightsideofmaths]

Glad you enjoyed it!

[@PSMathVideos]

Thanks you. Your videos have helped me so much in my classes.

[@brightsideofmaths]

You are so welcome! :)

[@ДмитроДмитренко-с7н]

I was struggling with Borel sets for half of a year. Thanks for giving me a good understanding)

[@M.H-91]

Let Ω be anonempty set and A⊆ B ⊆P(Ω). Then ℴ ⊆ ℴ

[@98raja]

Brilliant ! I loved this first lecture and I mean to view all the others. You are a great teacher. Well done from Thailand.

[@normansabin8322]

Bro, this is such a good job. Thank you.

[@adilmts3793]

thnx alot for your courses they are verry clear and dynamics than classical ones

[@rounakhosnijahan6958]

thanks a lot sir for giving such simple and neat explaination...it will be helpful for many students...you guys are the ones who make mathematics interesting and understandable...keep going sir, we want more videos regarding various topics of higher mathematis

[@owentong7319]

What used to confuse me is that when we say M and M^c, I used to think of putting M as a whole set (i.e a set of set) into the sigma-algebra.......... Good job!

[@mohaghaderi6077]

5:35 Sigma algebra must contain M, meaning {a, b}. Why does it need to contain {a} and {b} singletons though? Thanks for the videos.

[@brightsideofmaths]

Be careful: M is not the the set {a, b}. Here, M is given as the set that has the elements {a} and {b}.

[@mohaghaderi6077]

@@brightsideofmaths thank you!

[@rodilrun9195]

Because of your teaching ,I love mathematics

[@SingoStudios]

Thank you for these videos!

[@AakarshNair]

Congrats, this is really well done.

[@brightsideofmaths]

Thank you! Cheers!

[@mattcaruana6591]

does the set {a,b,c,d} need to be included in sigma(M)? Since it is the union of {a,b} and {c,d}

[@norbertdabrowski9319]

there is - X

[@jack_papel]

@@norbertdabrowski9319 Omg thank you for saying that.. I was scratching my head I was confused where it was

[@VamsiKrishna-fv5gg]

@@norbertdabrowski9319 Thanks for saying that man!! i literally scratched my head for so long and then saw your comment and felt like " how stupid of me!"

[@strai5150]

Thanks for the english version😊 ..hope you will upload more videos on english version😊

[@brightsideofmaths]

You are welcome. I am working on all lot of English versions. They will come :)

[@strai5150]

@@brightsideofmaths ..eagerly waiting for it😊😊

[@dijkstra4678]

At first I was really confused as to what a smallest sigma algebra of this thing M could mean but after your explanation it was very clear thank you!

[@brightsideofmaths]

You're very welcome!

[@davidyang977]

I would fail my probability course without you. Thank you friend :)

[@PunmasterSTP]

How did the rest of your class go?

[@aceofshade]

Hello, I have 2 questions 1. What is the significance of having a sigma-algebra in measure theory. Thank you for the great explanation on what is it but I am curious why it's so important to be taught first and how it helps us when we have these properties. Is it so that we can be sure that everything in the sigma-algebra is measurable? 2. What are examples of a subset of the real numbers that should not be in a Borel sigma-algebra. I guess I am confused why we "can't" have every subset since I assume the Borel sigma-algebra is already infinite in size.

[@brightsideofmaths]

Good question! I would suggest to watch the next 3 videos and then ask again :)

[@67254215415413]

Great video! I do have a question. You mention that not all subsets of an arbitrary X can be measurable sets, such as for the real number line. But then how do we guarantee that there even exists a sigma-algebra that contains M? Are all members of M required to be measurable sets?

[@brightsideofmaths]

You could choose the power set as a sigma algebra.

[@tlli5066]

My understanding about probability and measure theory was the measure of a null set. Now It has turned other way around, because of ur video

[@ativjoshi1049]

Neat, clear and concise tutorials. Can you share the notes as PDF ?

[@giomil98]

You are a true blessing! Thank you very much for the great content!!

[@brightsideofmaths]

I appreciate that! Thank you :)

[@HappyManStudiosTV]

beautiful explanation

[@TheXorion]

this is sooo much better than the book I am reading by alan karr.

[@VictorHugo-xn9jz]

For the "easy to show" part, how do you deal with the case where the index set is uncountable? Because taking the complement of the countable union of the complements of each measurable set gives the countable intersection, but not uncountable intersection.

[@brightsideofmaths]

Maybe you are confused by the "levels" here. We have an intersection of sigma algebras here, not an intersection of the subsets of X.

[@navjotsingh2251]

Do you plan on doing measure theoretic probability theory? That would be a nice topic to learn after covering measure theory.

[@brightsideofmaths]

Yes, indeed. I have this on my list :)

[@medha277]

At 2:04 I think you mean intersection instead of union

[@brightsideofmaths]

Yes :)

[@nnekaukwuegbu1170]

Thank you so much for this video. To illustrate with another example, if X = { 70,80,70,110}, am I correct to say that my sigma algebra is {empty set, X, {70,80,110}, {80,110}, {70,110}, {70,80}}?

[@rickmcn1986]

You need to include the singletons too? {80,110}^c= {70} isnt in there for example...

[@HarpreetSingh-ke2zk]

Help me, please. A_1, . . . , A_k are disjoint sets in B[0, ∞) × B (R\{0}), where B is a Borel set and x is the cartesian product. How do we interpret B[0, ∞) × B (R\{0})?

[@devaiyer9040]

Thank you!

[@munshikhan2265]

very helpful video on this topic

[@brightsideofmaths]

Glad it was helpful!

[@PrimevalShrimp]

Actually the sigma-algebra of M does not contain M, but the elements of M, which is a difference, as otherwise it would be a more nested set.

[@brightsideofmaths]

Edit: If you are referring to the generated sigma-algebra: As given in the formula, "containing" is here to be understood as a subset relation.

[@rasoolhasan2089]

Very good.I understood.thank you

[@colinsnyder1033]

We don't actually use the definition, that the smallest sigma algebra containing M is formed by taking the intersection of all sigma algebra's containing M, in the example following it 4:39, right? It seems to me we just construct it ourselves, is the definition used implicitly somehow?

[@brightsideofmaths]

The construction leads to the definition :)

[@susannduku243]

Thank you

[@brightsideofmaths]

Welcome!

[@farzanaafroz8661]

Please make more!!!

[@lemsolaris67]

Regarding condition b: why do you pose it as a condition ? If A is a subset of fancy A, how can it’s complent not be what you pose it as a condition?

[@brightsideofmaths]

Take fancy A = { B }. Then B^c is not in fancy A.

[@Jacky_Wang-wlm]

Thank you！

[@brightsideofmaths]

Thank you for your support!

[@randomgirl7000]

at 8.13 don't we have to add {a,b,c,d} to the smallest algebra since {a,b} and {c,d} are elemnts of the algbera then their union should be an element of sigma algebra

[@brightsideofmaths]

The whole set (=X) has to be included, yes!

[@ryanlangman4266]

Does a sigma algebra over X always also form a topology over X. My reasoning is that: X and {} must be in the sigma algebra, the infinite union of any elements in the sigma algebra is in the sigma algebra, and the finite intersection of any 2 elements in the sigma algebra must be in the sigma algebra. Proof: The first 2 are trivially equivalent to the axioms of sigma algebras. Let A,B be elements of the sigma algebra, Then A^c, B^c are also in the sigma algebra, So A^c U B^c and (A^c U B^c)^c are in the sigma algebra, So, A intersect B is the the sigma algebra. Therefore the sigma algebra forms a topology over X.

[@martinpuente7526]

hi! why did you add the complement of {a,b} in 6:45? bc the second condition is M\X... that's just {b,c,d} and {a,c,d}... it's still a sigma algebra but I don't know why you add it. Sorry for being so annoying but I want to learn it well. I really liked your channel

[@brightsideofmaths]

We have to add all the unions and then the all the complements as well.

[@jsusss]

Great! Thanks for this pill 😊

[@oronamath-phs7913]

I am quite new to this, how would we prove the 'easy to show' part (if Ai is a sigma algebra on X then the intersection of Ai's is also a sigma algebra on X)? Thank you.

[@brightsideofmaths]

Hello and welcome to the club! Everytime, I say "easy to show", it does not mean that it is simple, straightforward, short, or immediately given. It just means that, after understanding the topic, one can get the correct idea to write down the proof. It is the same here. Just take some paper, write down the properties of a sigma-algebra and then what you need to prove.

[@rocktimjyotidas4543]

I did not understand the part about borel sigma algebra?

[@dmitry.i.kabanov]

When you say at 2:20, "For M \subseteq P(X), there is the smallest sigma-algebra that contains M", then it is a bit confusing because the sigma-algebra contains sets from M, not M itself.

[@brightsideofmaths]

However, M is a subset of the sigma-algebra. This is what I meant with "contain" here. Of course, I agree, all this can be very confusing :)

[@dmitry.i.kabanov]

@@brightsideofmaths Ah, I see, I was thinking you meant $M \in \sigma$. If it means \subseteq, then I agree.

[@nerdemoji280]

Thank you very much for these video series. I am a little confused on something. Are we defining the topology as a set of the open subsets in X? In which case, we know that the topology is a subset of the power set. However, when I look it up, the topological space is defined as a family of subsets. Is this the same thing? In the case I said above, I am getting stuck on the part where you say that the Borel sigma algebra is the sigma algebra generated by the open sets. I am assuming that it means the sigma algebra generated by the topology (or equivalently, the sigma algebra generated by the set of open sets). Is this correct? And just to confirm, the topology and the sigma algebra are separate subsets of the power set of X, correct?

[@brightsideofmaths]

The topology is defined by saying which sets should be called "open". So a topology is a collection of subsets, yes! In the same way, a sigma-algebra is a collection of subsets. However, both collections satisfy different rules, so in general there are not the same.

[@PunmasterSTP]

@@brightsideofmaths Thanks; your comment cleared things up for me as well!

[@sylviamalik1030]

What are the open sets on R?

[@demerion]

The way you created the example at 4:40 was not by using the definition above, right? Because we didn't look at intersections? Or did we do that but it wasn't obvious?

[@brightsideofmaths]

Indeed, we didn't need to use the definition in the formal way because we immediately have seen how the smallest sigma-algebra has to look like.

[@demerion]

@@brightsideofmaths Okay, vielen Dank! An actual example would have been great, but I guess there are examples out there, so it's not that big of an issue :)

[@brightsideofmaths]

@@demerion It is not like one would actually calculate the intersection for an example. It is just a good thing to work with this definition in the general case.

[@andregormann3781]

7:45 The sigma-algebra is missing {a, b, c, d}.

[@brightsideofmaths]

And what is X? ;)

[@tsunningwah3471]

Stay healthy💪🏻

[@Krahltan]

8:20. Sigma(M) doesn't look like a sigma-algebra. It fails to satisfy the 3rd property of sigma-algebras, unions. i.e. {a} U {b,c,d} = {a,b,c,d} which is not in sigma(M). So Sigma(M) should be P(X) / {{a,c},{a,d},{b,c},{b,d},{c},{d}}. Am I missing something?

[@garrycotton7094]

{a,b,c,d} is X itself which is included :)

[@PrimevalShrimp]

I think the (c) property of sigma-algebras is wrong, as it is not defined for the union of all indices i from 1 to infinity, but just for arbitrary i in N, like U (i in N).

[@brightsideofmaths]

Implicitly, I assumed that you read (c) in the way: If for all i in N, A_i lies in A then: ...

[@jacobguerreso675]

What's wrong with the captions? Might be a glitch but CC and transcript is giving what was in "part one" of the series.

[@brightsideofmaths]

How is it now?

[@jacobguerreso675]

@@brightsideofmaths Looks good! May have just been a glitch on my end. Thank you

[@brightsideofmaths]

Thanks! I just uploaded the correct one again. I don't know what caused the error but I am very glad that people tell me about these things :)

[@PunmasterSTP]

Q: What do you call it when a student flips over having to learn about sigma algebras? A: A Borel roll! 😎

[@thesilentvoice3397]

Could tell me which books u refer for measure theory, as after watching this it would be easy to get along with concepts in books

[@brightsideofmaths]

There are a lot of good books in measure theory. I really like Schilling's "Measures, Integrals and Martingales". However, books are always a matter of taste. Just test some of those in the library before buying.

[@prajeesharoseroyson7809]

Very easy to understand 🥰

[@brightsideofmaths]

Thanks :)

[@raiza.chakufora5271]

Excellent explanation. How do we prove that uncountable subsets of R is not a borel sigma set. Pliz help me understand this

[@brightsideofmaths]

R is an uncountable subset of R and a Borel set.

[@raiza.chakufora5271]

@@brightsideofmaths I can't see the prove I requested to understand. Prove that uncountable subset of R is not a borel sigma algebra

[@brightsideofmaths]

@@raiza.chakufora5271 There is no proof for that because it's not a true statement :D

[@raiza.chakufora5271]

@@brightsideofmaths Really!

[@user-dq8uc5gv2u]

Why is it important that we're chosing only open sets when creating our Borel Sigma Algebra? Why is the open sets the ingredient we need from the topological space?

[@brightsideofmaths]

The topological space is completely determined by the open sets. Therefore we can extract this information by considering these open sets and form our sigma-algebra.

[@mariamoralescruz5498]

this is so amazing!!! thanks god ur here to save all of us :)))

[@joshuaokoro2781]

What is Borel measurability?

[@ryanhutchins2634]

What program/tools do you use to make these videos? I think I would like to follow your form.

[@brightsideofmaths]

Glad you like it! Which topics do you want to cover? :) I use the nice free program Xournal!

[@ryanhutchins2634]

@@brightsideofmaths I was considering starting by bridging the material that you cover in your measure theory series to specific, computational examples from the theory of probability. I'd like to explain the "full stack" of proabilistic computations from sigma algebras and measures up to Riemann integral computations in probability. As part of that, I might also show proofs of deep truths in probability with measure theory side-by-side with the way they would look in a more computationally focused course, or a beginning probability class without measure theory, and illustrate the connections (because I want to understand them myself!). I'm only on lecture 6 in your series right now, so I'm not sure if you delve into those connections, but I am currently in Wahrschenlichkeitstheorie I at Uni Heidelberg, and I recently taught myself the beginning course in probability from Prof. Blitzstein's lectures (on KZbin) from Stats 110 at Harvard. I love your videos for giving me another perspective on the important parts of the material. Also, they're in English, which is a bit helpful. I fear I sometimes miss some of the very subtle details when I read the German Maßtheorie textbook and listen to the lectures at the Uni, although I know I'm improving. Apologies for the terribly long answer.

[@markpayton3895]

I don't get why the original X is required to be part of the sigma algebra ??? For example, if we are trying to find the sigma algebra of a subset (i.e. M) of the real line, how can it be that the entire real line (i.e. X) is also a sigma algebra ???

[@brightsideofmaths]

A sigma algebra is a collection of subsets! So the real number line is not a sigma algebra.

[@ichkaodko7020]

I have a question regarding to sigma algebra. According to def of a. It must have empty and full set, then my question is if full set is included, then why the elements of full set X is not used in the example? I mean, in the example, sigma(M)'s elements all above mentioned elements and not single element from full set X.

[@brightsideofmaths]

I don't understand, X is explicitly included in the example.

[@ichkaodko7020]

@@brightsideofmaths basically, I am struggling with idea of X is explicitly included in the example but even if X is included, u don't write or use its elements explicitly. I thought explicitly including X means we must use X's elements as well?

[@brightsideofmaths]

@@ichkaodko7020 Please don't forget that all elements of the sigma algebra have to be sets!

[@ichkaodko7020]

@@brightsideofmaths ah, i see that's why we don't include the elements of X hence it is not set. I got it now. Thank you very much. I am so grateful that you always answer my question.

[@brightsideofmaths]

@@ichkaodko7020 You are very welcome :)

[@Wavams]

at 2:03 you say union, but you mean intersection right?

[@brightsideofmaths]

Opps! You are completely right! Sorry for the confusion!

[@Wavams]

@@brightsideofmaths no problem, thanks for the very clear videos!

[@brightsideofmaths]

@@Wavams Thank you very much. However, you see, some mistakes can always happen ;)

[@lucasjones5445]

Wait - isn't it supposed to be unions? Unions are again used at 5:53. Wiki says the third property is "closed under countable unions". So the symbol used in the video is also incorrect? Sorry if I'm wrong.

[@gillgousse2247]

@@lucasjones5445 Hi, I don't know if you figured it out but just in case; the symbol is correct, he's indeed speaking about intersections. One of the axiom of a sigma-algebra is closure under countable unions, that's right. However, here he's talking about intersections of sigma-algebras. Countable intersection of sigma-algebras on a set X is a sigma-algebra on X. This is not necessarily the case when you take unions of sigma-algebras.

[@shibo1038]

hi, where can we find the notes?

[@hamidelll2850]

Sigma algebra union of {a,b} {c,d} is {a,b,c,d} Think you after all

[@brightsideofmaths]

What do you mean by "sigma algebra union"?

[@jyotiupadhyaya9333]

Very nice🎉

[@asht750]

Correction: The Borel s-algebra is the smallest s-algebra generated by the open sets in R.

[@brightsideofmaths]

And it's also the largest sigma-algebra generated by the open sets. The attribute "smallest" does not give more information here :)

[@asht750]

Yes!@@brightsideofmaths Does any collection of open sets in R form a Borel sigma algebra?

[@therasmataz2168]

@ 2:54 I"m confused please help, you claim M does not need to be a sigma algebra yet can form sigma algebras that contain M??!

[@therasmataz2168]

Ah I see it now; You created the smallest sigma algebra of M by including the empty set, X, M and the appropriate unions/intersections. I look silly now but you sir are a great expositor of mathematics. Thank you so much ☺

[@brightsideofmaths]

Thanks a lot :)

[@tag_of_frank]

What is and what is not in the borel sigma algebra, compare it to R. The video is titled Borel Sigma Algebra, yet 2 minutes out of 12 are discussing it, and barely scratch the surface

[@brightsideofmaths]

Yeah, watch the following videos.

[@err3088]

GOLD

[@haggaisimon7748]

2:10 not union but intersection.

[@brightsideofmaths]

Correct. My mistake! Thanks!

[@haggaisimon7748]

@@brightsideofmaths no problem. I like your series. Much needed.

[@rafaelschipiura9865]

Doesn't make sense to me. The definition of the σ-algebra requires a countable amount of sets in it's definition. But then you jump to a σ-algebra formed from a set with uncountable many sets in it.

[@brightsideofmaths]

No, this is not correct. The definition of the sigma-algebra does not require countably many sets. On the contrary, most interesting sigma-algebra have uncountably many set.

[@rafaelschipiura9865]

@@brightsideofmaths I'm sorry I didn't explain myself very clearly. I know σ-algebras can have uncountably many sets in them, for example 𝒫(ℕ) is uncountably infinite. What I don't see is how one goes from the definition, which talks about the union of a number of countable sets (possibly countably infinite) to such σ-algebras that have uncountably infinite sets on them.

[@rafaelschipiura9865]

@@brightsideofmaths Also, thank you very much for your time, these lectures are wonderful.

[@rafaelschipiura9865]

@@brightsideofmaths I think I get it now. A σ-algebra can contain uncountably infinite sets. To be a σ-algebra it's not required that it contains the union of uncountably infinite sets, just that it contains the union of all countable subsets of them.

[@brightsideofmaths]

@@rafaelschipiura9865 Perfect!

[@jayjayf9699]

I still dont understand borel sigma algebra

[@brightsideofmaths]

What is exactly your problem with it? The definition?

[@jayjayf9699]

@@brightsideofmaths just the concept of measure theory seems too abstract for me to comprehend, i am using your videos, plus lecture notes from various universities, i am self teching my self, i cant even attempt the problem sets of these university exercise sheets, im sure you will get alot more of my comments, i understand now that the borel sigma algebra is the smallest generated sigma algebra that is generated by all the open sets of the real line?

[@yaoweizhang8354]

ich finde deinen Akzent ziemlich suess

[@lorenzoruggeri_]

sei mio padre, un grande

[@sandeepkumarsaw3109]

(a, c, d) is not in the sigma

[@brightsideofmaths]

What do you mean?

[@hamidelll2850]

9 elements in sigma algebra ,not 8 elements

[@brightsideofmaths]

Is this a riddle? :D

[@hamidelll2850]

@@brightsideofmaths no I m wrong

[@jayjayf9699]

I understand the sigma algebra generated by M, but how did that link to borel sigma algebra

[@brightsideofmaths]

The Borel-Sigma-Algebra is just a notation for this generated sigma algebra.

[@aijaznazir2648]

Nice content i have also started my KZbin channel providing videos on topology functional analysis algebra

[@seka8601]

Well but way too slw explained

[@brightsideofmaths]

Thanks! You have a point but don't forget that KZbin offers faster replays :)

[@MrArmas555]

++

[@rikterr]

ma come parla sto tizio