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! th-cam.com/video/xZ69KEg7ccU/w-d-xo.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.
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!
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 literately spent 12 mins on TH-cam 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. :)
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
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 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.
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 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.
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!
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!
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.
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.
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 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.
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?
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.
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?
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...
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?
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).
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.
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!
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.
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 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!
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?
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.
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 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.
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! th-cam.com/video/xZ69KEg7ccU/w-d-xo.html
"Page not found" ?
@@NilodeRoock I've updated the link now :)
@@brightsideofmaths Thank you.
Thank you very much sir
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
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.
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!
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.
I literately spent 12 mins on TH-cam 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.
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
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.
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
Now after watching this
I can say that measure theory is measurable 😅
Thanks for this wonderful video ❤️
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.
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.
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!
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! :)
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 😀.
You are the professor of MIT level, you video lectures should be accepted, respected, appreciated and advocated!!!!!!!!
Tried to understand this from a book and didn't. This video enabled me to grasp this easily. Great video!
Glad it was helpful!
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.
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.
Amazing video! Amazing series! Please keep it coming! Measure theory has never been easier to understand. Thank you!!
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!
Thank you, straight after the lecture I watch your lectures.
Best idea :)
Even more succinct and concise than my lectures but I understand it a lot more. Wow. Thank you!
These videos are amazing and incredibly helpful!! Thank you SO much!!
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
This video helped me a lot, thank you!
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
I love to watch your videos to get notion about the subject before reading handbook. Great job !
Thanks :)
Is X here the same as large Omega (sample space) ? Since X is the complement of the empty set. Thx.
Yes, exactly :)
Thanks for taking the time to produce this content, it brought me back memories when I was studying this course.
Thank you so much for all videos. Your teaching skill is amazing.
Just starting measure-therotic probability theory and these are great :)
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...
The moment I realised this dude is giving a brief explanation on Measure Theory, I subscribed immediately.
Best measure theory video on YT!!!!
Glad it was helpful!
Clearly explained, basic examples, very good
This channel is amazing! So glad i found it! I subscribed of course
This was incredibly helpful, thanks for the knowledge!
My man, you kinda sound like Mimir from God of War. Loving it!
Very well explained with straightforward and intuitive examples.
We neeeeeeeeed more of this exciting course in Mathematics.
Keep it up~!!!
The presentation was amazing. Thank you!
Glad you liked it!
I finally understand why a sigma algebra is the way it is. The drawing made it so clear to me
Nice :)
Very intuitive explanation, thank you. Very helpful for Engineers 👍
Glad it was helpful! :) And thanks for the support!
Thank you.
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!
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.
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 goodness, you really do have a video in everything I'm looking for.
Thank you very much :)
The nomenclature - 'sigma-algebra' etc - is far more fearsome than the reality!
Indeed :D
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?
I am a sigma algebra male
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.
This is actually a great explanation.
Greetings from Spain!
Beautiful insight of the topic
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.
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 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.
I love your work man
Glad you enjoy it! And thanks for your support :)
Beni buraya kadar getiren eğitim sistemimize teşekkür ediyorum.
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...
So glad I've found your channel. Which book did you use to study this?
I went through this crap in introduction to probability and was totally lost. Thank you for explaining.
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
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.
Thanks so much sir.This is amazing and helpful
Thank you so much. now finally i can understand it.
Great explanation. Kudos!
Love you’re videos
You are one of the best teachers I’ve had
Came here for functional analysis, stayed for measure theory
When I click on like, nothing happens but if I click on dislike it says comment shared with... Thanks for the video. It's pretty clear
Don't hit dislike :D
Excellent video lecture
Thank you very much.
Thank you, it was a very helpful video!
Very useful. Thank you sir.
Thank you 🙏🙏🙏🙏 I finally understood
You are very welcome :)
Very insightful explanations. Some people are born lucky!
The name is so scary, so we need people like you in this world to make them look less intimidating, thanks for the explanation
Sigma algebra for Sigma mathematics
Loved it, great video!
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!
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 :)
Is there a book you'd recommend to match this content?
You can try out Schilling's book!
This is the first video where i understand what a sigma algebra is, thank you!
Thank you!
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.
Fantastic work!
Thank you this helped so much!
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).
thank you really I enjoy these topics
THIS VIDEO IS AMAZING!!
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!
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 :)
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
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
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.
Just awesome. Loved it!
Great video, man!