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! ruclips.net/video/xZ69KEg7ccU/видео.html
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!
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.
I literately spent 12 mins on RUclips 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. :)
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!
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.
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 :)
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
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.
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.
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!
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!
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.
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.
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!
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.
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.
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.
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.
@@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.
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...
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
@@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.
@@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.
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?
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).
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?
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.
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.
Your videos are amazing! Really! I would like to know what software do you use to have that yellow screen and that logo, I would like to use that to my own courses in economics! Thank you.
@@brightsideofmaths Thank you! Would you also share what notetaking equipment and microphone do you use? I use Wacom bamboo and a regular mic, but I'm not sure it is good enough.
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 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.
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.
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
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).
@@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
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)?
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.
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...
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!
Hi,I followed your lectures on measure theory and really enjoyed it,I have my interview on this topic and I want to provide the refference of book I used for my presentation,could you please tell me which book you followed?like I do not find the general definition of measurable functions in any book provided by you i.e a function from one measurable space to other is said to be measurable if inverse image of measurable set is measurable,....even though I saw this definition many times on online ......please just provide book and author name so that I feel confident in providing refferences if required,thanks 😊
Thank you very much. You can reference my actual name and my videos without any problems because I am a real person and a real mathematician. :) If you want to cite a book, I would to go with Schilling's MEASURES, INTEGRALS AND MARTINGALES.
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).
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).
Hi, thank you so much for posting this series! It helps me a lot with my analysis course! Also I have a question here: since the power set of a set X is always sigma-algebra, does that mean any subset which belongs to the power set is measurable with regard to the power set?
The third property that the sigma algebra is closed under countable unions, is it the same as saying any two events that belong in the collection will have their unions in the collection as well for eg say elements A1 and A8 of the sequence, can one say that their union is also an element of the sigma algebra or is it not ? also can we not index the collection in any arbitrary way we want and if the union does not start with i = 1, then their final unions will all be different. So is the criteria demanding all possible unions of all possible sequences to also be members ? This has been bugging me for a while. Thanks in advance !
The key word is "countable". This is the smallest infinity. We demand that the infinite union stays in the sigma-algebra. However, it still needs to be countable, which means indexed by the natural numbers.
Thank you for the video. I found it very informative. Can you suggest some good books for studying basic measure theory? I have taken real analysis and complex analysis courses if that helps. Thank you.
I would suggest Folland, "Real Analysis: Modern Techniques and Their Applications". Also Rudin's "Principles of Mathematical Analysis" has some material on the Lebesgue integral. For a more advanced treatment, see Rudin's "Real and Complex Analysis". Finally, Fristedt and Gray, "A Modern Approach to Probability Theory", covers measure theory from a probability viewpoint.
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!
Just a small criticism, if you want it. A little confusing because your P(A) notation for the power set isn't fancy and it looks like probability of A.
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?
Hi, I am wondering how is the measure theory connected with the distribution theory? We have introduced with the correspondence theorem when having a measure-based probability class. But I didn't quite get the idea...
Good point. Distribution theory has a lot of connections to measure theory. When you see the dirac measure, one immediately recognise this. However, at the moment, I don't have any videos about this connection. They will come for sure :)
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)
Actually, I am not following any particular textbook. However, if you want some recommendations, I can tell you that Schilling's book and Elstrodt's book are very good!
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! ruclips.net/video/xZ69KEg7ccU/видео.html
"Page not found" ?
@@NilodeRoock I've updated the link now :)
@@brightsideofmaths Thank you.
Thank you very much sir
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!
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.
What you are doing is amazing. I hope you can produce more content in English for non-German speakers.
I honestly feel like learning German to get access to mroe videos
I literately spent 12 mins on RUclips 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. :)
Thanks :)
I have the same experience as yours.
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!
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.
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 :)
I bet when he wrote that he was thinking about topology.
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
You are the one who really can make students as well as teachers to understand measure theory in real meanings
Thank you very much :)
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.
I'll come back to this video when I'm stronger. Need more training.
Just farm some EXP on my lower level videos ;)
buy more pots
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.
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!
Wow, thank you! :)
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!
Tried to understand this from a book and didn't. This video enabled me to grasp this easily. Great video!
Glad it was helpful!
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.
does this mean that a sigma-algebra is a group under union?
@@axelperezmachado5008 It isn't because there is no inverse of union in the sigma algebra
@@deept3215 True! Haven't realised
@@axelperezmachado5008 It's an abelian group with respect to the operation of symmetric difference.
You are the professor of MIT level, you video lectures should be accepted, respected, appreciated and advocated!!!!!!!!
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.
Now after watching this
I can say that measure theory is measurable 😅
Thanks for this wonderful video ❤️
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!
The moment I realised this dude is giving a brief explanation on Measure Theory, I subscribed immediately.
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.
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.
@@sheerrmaan Exactly.
Extremely marvelous explanation really enjoyed it
Please also upload lectures of complex analysis
Already done. See here: tbsom.de/s/ca
Thanks sir
Even more succinct and concise than my lectures but I understand it a lot more. Wow. Thank you!
Amazing video! Amazing series! Please keep it coming! Measure theory has never been easier to understand. Thank you!!
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 😀.
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.
You've never understood it because nonsense cannot be understood, only believed.
Would you ever consider making a maths series on the subject of topology? Your videos are brilliant!
Thanks! I want to do that, yes :)
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.
@@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.
Best measure theory video on YT!!!!
Glad it was helpful!
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...
From India a great respect for you . Your videos are amazing
Glad you like them! :) And thanks for your support!
at 08:15 can you explain what do you mean by countable union of infinitely many sets ?
The index set for the union is given by the natural numbers :)
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
Thank you! I don't understand your question completely. Can you elaborate on that?
@@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.
Answer: Yes, it's possible but uninteresting ;)@@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.
@@Hold_it yes! that was exactly what I meant thanks
Is X here the same as large Omega (sample space) ? Since X is the complement of the empty set. Thx.
Yes, exactly :)
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 ?
A is an element of P(X), but a subset of X.
@@brightsideofmaths thank you so much , you are amazing ;)
Thanks for taking the time to produce this content, it brought me back memories when I was studying this course.
I finally understand why a sigma algebra is the way it is. The drawing made it so clear to me
Nice :)
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?
Thank you, straight after the lecture I watch your lectures.
Best idea :)
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).
You are totally right! I often used redundancy is definitions to make them clearer.
This is actually a great explanation.
Greetings from Spain!
I love to watch your videos to get notion about the subject before reading handbook. Great job !
Thanks :)
Clearly explained, basic examples, very good
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?
There is no difference. Both sets P(X) from you are exactly the same.
So glad I've found your channel. Which book did you use to study this?
The name is so scary, so we need people like you in this world to make them look less intimidating, thanks for the explanation
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.
A is a part of X and an element of P(X). So these are the same pictures but with different visualizations.
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.
You are one of the best teachers I’ve had
Came here for functional analysis, stayed for measure theory
Very well explained with straightforward and intuitive examples.
We neeeeeeeeed more of this exciting course in Mathematics.
Keep it up~!!!
My goodness, you really do have a video in everything I'm looking for.
Thank you very much :)
Your videos are amazing! Really! I would like to know what software do you use to have that yellow screen and that logo, I would like to use that to my own courses in economics! Thank you.
I uses the wonderful old program Xournal.
@@brightsideofmaths Thank you! Would you also share what notetaking equipment and microphone do you use? I use Wacom bamboo and a regular mic, but I'm not sure it is good enough.
@@caio868 A Wacom is great. Your mic, you can just test. Maybe the audio quality is already sufficient. Have fun!
Thank you so much here! I have an exam in a few days and you're literally saving me :)
Happy to help! :) And thanks for the support!
My man, you kinda sound like Mimir from God of War. Loving it!
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!
Very intuitive explanation, thank you. Very helpful for Engineers 👍
Glad it was helpful! :) And thanks for the support!
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.
Thank you! I also want to do quizzes for the other parts if they are helpful.
@@brightsideofmaths They are helpful for retention of material and application, IMHO. Math is not a spectator sport, IMHO.
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.
Thank you! What exactly is wrong about the Venn diagram?
@@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.
Is there a book you'd recommend to match this content?
You can try out Schilling's book!
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.
What is your book?
@@brightsideofmaths capinski and kopp: measure integral and probably. You bring this into effect in a later video.
I went through this crap in introduction to probability and was totally lost. Thank you for explaining.
This is such a helpful video! Now i feel like i can pass measure theory
What course are u taking this for? 🤔
@@boyzrulethawld1 id guess measure theory
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
There, you are completely right! However, I really like this definition because one sees the smallest Sigma-Algebra immediately in the first part.
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).
@@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
These videos are amazing and incredibly helpful!! Thank you SO much!!
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)?
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.
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...
Just starting measure-therotic probability theory and these are great :)
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!
But X is a subset of X as well :)
which board you are using? I am also interested in making tutorial videos but unable to find the board like yours!
I am using a simple Wacom board :)
@@brightsideofmathsi mean the software☺
@@kusalthapa3570 Xournal :)
@@brightsideofmaths thanks❤️
I have a question. At 10:18 do you mean every set in sigma-algebra is measurable set???
By definition: We *call* the elements of a sigma-algebra _measurable_ .
@@brightsideofmaths thanks
Hi,I followed your lectures on measure theory and really enjoyed it,I have my interview on this topic and I want to provide the refference of book I used for my presentation,could you please tell me which book you followed?like I do not find the general definition of measurable functions in any book provided by you i.e a function from one measurable space to other is said to be measurable if inverse image of measurable set is measurable,....even though I saw this definition many times on online ......please just provide book and author name so that I feel confident in providing refferences if required,thanks 😊
Thank you very much. You can reference my actual name and my videos without any problems because I am a real person and a real mathematician. :)
If you want to cite a book, I would to go with Schilling's MEASURES, INTEGRALS AND MARTINGALES.
@@brightsideofmaths what is your name?
@@ibrahimislam8759 You can find that in my about page on RUclips, where there is also a "impressum" linked.
Just go here :)
ruclips.net/user/brightsideofmathsabout
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!
Is this series enough to understand the measure theory in a good sense or did it finished?
Yes, it's for understanding measure theory :) I also have additional series to deepen your knowledge, see description.
@@brightsideofmaths thank you so much. I ve never learnt measure theory. Does it have challenges like hard problems?
This channel is amazing! So glad i found it! I subscribed of course
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).
Hard to answer. There are a lot of good books about measure theory. Do you know Bauer?
@@brightsideofmaths Yes! Are you following Measure and Integration Theory by Heinz Bauer?
No, but I recommend the book :)@@sarthakgupta1165
Thank you so much for all videos. Your teaching skill is amazing.
Do the subsets A_i always need to be disjoint, like you drew it, or is it not necessary according to this definition?
We don't assume disjoint sets in this definition :)
@@brightsideofmaths ok great, thank you 🙌🏼
This video helped me a lot, thank you!
Ideally one starts with this after some calculus and modest exposure to probability topics, am I correct? Trying to construct a curriculum for myself.
I have a website for this :) tbsom.de/startpage
@@brightsideofmaths Much appreciated!
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).
I have no idea what I am doing. I just graduated. But does these sets have something to do with that set paradox letter?
Hi, thank you so much for posting this series! It helps me a lot with my analysis course! Also I have a question here: since the power set of a set X is always sigma-algebra, does that mean any subset which belongs to the power set is measurable with regard to the power set?
Thank you very much for the question! You are completely correct :)
I would like to know what this whiteboard app is. Its look very clean and simple.
This is the first video where i understand what a sigma algebra is, thank you!
The third property that the sigma algebra is closed under countable unions, is it the same as saying any two events that belong in the collection will have their unions in the collection as well for eg say elements A1 and A8 of the sequence, can one say that their union is also an element of the sigma algebra or is it not ? also can we not index the collection in any arbitrary way we want and if the union does not start with i = 1, then their final unions will all be different. So is the criteria demanding all possible unions of all possible sequences to also be members ? This has been bugging me for a while. Thanks in advance !
The key word is "countable". This is the smallest infinity. We demand that the infinite union stays in the sigma-algebra. However, it still needs to be countable, which means indexed by the natural numbers.
@@brightsideofmaths so in the video Ai can be any sequence of sets in the sigma algebra
@@delq Exactly.
@@brightsideofmaths Thank you ! ✌
Could you make more videos about the hysteresis system that described by measure theory
Sounds like a very good idea. Do you have more details there what you want?
The presentation was amazing. Thank you!
Glad you liked it!
Thank you for the video. I found it very informative. Can you suggest some good books for studying basic measure theory? I have taken real analysis and complex analysis courses if that helps. Thank you.
I would suggest Folland, "Real Analysis: Modern Techniques and Their Applications". Also Rudin's "Principles of Mathematical Analysis" has some material on the Lebesgue integral. For a more advanced treatment, see Rudin's "Real and Complex Analysis". Finally, Fristedt and Gray, "A Modern Approach to Probability Theory", covers measure theory from a probability viewpoint.
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!
Just a small criticism, if you want it. A little confusing because your P(A) notation for the power set isn't fancy and it looks like probability of A.
Yes, there are always overlaps in notations in mathematics. So it is always important to put it into the context.
isn't (c) trivially true for any set? should it be the union of all the subset is equal to the set itself?
No, we are talking about a collection of sets :)
@@brightsideofmaths ohhh, i'm dumb
Thanks for this video. I have a question: like metric spaces, are measure spaces also useful in Fixed Point Theory? Thanks in advance
Yes, a lot of metric spaces are given with the help of measure spaces.
Very insightful explanations. Some people are born lucky!
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?
Probability = 0 does not mean "impossible". I have a whole video series about that :)
tbsom.de/s/pt
Hi, I am wondering how is the measure theory connected with the distribution theory? We have introduced with the correspondence theorem when having a measure-based probability class. But I didn't quite get the idea...
Good point. Distribution theory has a lot of connections to measure theory. When you see the dirac measure, one immediately recognise this. However, at the moment, I don't have any videos about this connection. They will come for sure :)
Beautiful insight of the topic
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)
thank you for sharing this amazing video! definitely love it
Thank you for your support :)
Hi thank you for sharing. May I know which textbook are you following? Thank you
Actually, I am not following any particular textbook. However, if you want some recommendations, I can tell you that Schilling's book and Elstrodt's book are very good!
This was incredibly helpful, thanks for the knowledge!
How does Steady work? If you sign up is there a minimum time period to be a member?
You just sign up and can go anytime you want. In short the minimum time period is always a month.
@@brightsideofmaths Thank you!