Measure Theory 1 | Sigma Algebras
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- Опубліковано 30 тра 2024
- Find more here: tbsom.de/s/mt
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Watch the whole video series about Measure Theory and download PDF versions and quizzes: tbsom.de/s/mt
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This is my video series about Measure Theory. I hope that it will help everyone who wants to learn about it. We discuss sigma algebra, measures, and integration. For any questions, please leave a comment or come to the community forum of the Bright Side of Mathematics: tbsom.de/s/community
#MeasureTheory
#Analysis
#Integral
#Calculus
#Measures
#Mathematics
#Probability
This is part 1 of 22 videos.
00:00 Introduction
00:58 Measuring lengths
03:00 Example power set
03:51 Definition sigma-algebra
10:20 Example for sigma-algebras
(This explanation fits to lectures for students in their first year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)
The Bright Side of Mathematics has whole video courses about different topics and you can find them here tbsom.de/s/start
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! ua-cam.com/video/xZ69KEg7ccU/v-deo.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
I literately spent 12 mins on UA-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.
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!
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.
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.
Amazing video! Amazing series! Please keep it coming! Measure theory has never been easier to understand. Thank you!!
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.
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'll come back to this video when I'm stronger. Need more training.
Just farm some EXP on my lower level videos ;)
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Thanks for taking the time to produce this content, it brought me back memories when I was studying this course.
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!
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! :)
These videos are amazing and incredibly helpful!! Thank you SO much!!
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
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 😀.
Even more succinct and concise than my lectures but I understand it a lot more. Wow. Thank you!
Thank you so much for all videos. Your teaching skill is amazing.
This video helped me a lot, thank you!
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!
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.
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.
This was incredibly helpful, thanks for the knowledge!
This channel is amazing! So glad i found it! I subscribed of course
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!
Clearly explained, basic examples, very good
I love to watch your videos to get notion about the subject before reading handbook. Great job !
Thanks :)
Just starting measure-therotic probability theory and these are great :)
You are the one who really can make students as well as teachers to understand measure theory in real meanings
Thank you very much :)
Absolutely clear explanation.
Thank you, straight after the lecture I watch your lectures.
Best idea :)
Beautiful insight of the topic
Great explanation. Kudos!
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...
Now after watching this
I can say that measure theory is measurable 😅
Thanks for this wonderful video ❤️
The presentation was amazing. Thank you!
Glad you liked it!
Loved it, great video!
Fantastic work!
Thanks so much sir.This is amazing and helpful
So glad I've found your channel. Which book did you use to study this?
Thank you, it was a very helpful video!
Thank you this helped so much!
Excellent video lecture
This is amazing!! Thank you so much 🙏
This is actually a great explanation.
Greetings from Spain!
You are the professor of MIT level, you video lectures should be accepted, respected, appreciated and advocated!!!!!!!!
Great video, man!
Just awesome. Loved it!
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.
Very well explained with straightforward and intuitive examples.
We neeeeeeeeed more of this exciting course in Mathematics.
Keep it up~!!!
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.
Wow!! These explanations are really nice. Immediately subscribed 👌
What books can we refer to understand more about Measure theory, distribution functions, chebyshev lemme etc?
There are a lot of books. I really like Schilling's about measures and other stuff :)
@@brightsideofmaths Danke gut !
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
Very intuitive explanation, thank you. Very helpful for Engineers 👍
Glad it was helpful! :) And thanks for the support!
I love your work man
Glad you enjoy it! And thanks for your support :)
Extremely marvelous explanation really enjoyed it
Please also upload lectures of complex analysis
Already done. See here: tbsom.de/s/ca
Thanks sir
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.
thank you really I enjoy these topics
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!
Love you’re videos
Great video!
Very useful. Thank you sir.
My goodness, you really do have a video in everything I'm looking for.
Thank you very much :)
THIS VIDEO IS AMAZING!!
good job mate thanks a lot, subscribed
Thank you ! its such a good explanation.
Glad it was helpful!
Very insightful explanations. Some people are born lucky!
The moment I realised this dude is giving a brief explanation on Measure Theory, I subscribed immediately.
You are one of the best teachers I’ve had
Came here for functional analysis, stayed for measure theory
Thank you, really great work :)
Glad you liked it!
Thank you so much. now finally i can understand it.
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)
My man, you kinda sound like Mimir from God of War. Loving it!
Extremely clear and nice. Thank you so much! New subscriber here.
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 :)
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?
Man, you are a great teacher 👍
I appreciate that! Thanks :)
Best measure theory video on YT!!!!
Glad it was helpful!
Nice video!
Hello! Which recording software and writing software did you use for these? Love your channel by the way!
Xournal and HyperCam :)
The Bright Side Of Mathematics Thank you!
I would like to know what this whiteboard app is. Its look very clean and simple.
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.
I went through this crap in introduction to probability and was totally lost. Thank you for explaining.
Thank you very much.
Thank you 🙏🙏🙏🙏 I finally understood
You are very welcome :)
Thank you!
Thanks! I have one question regarding the bullet (c). Do subsets A_i have to be mutually exclusive, i.e. A_i intersection A_j = empty set? You draw them like that but you don't mention that this condition should hold for them.
No, not at all. The intersection does not have to be empty in condition (c). My picture was just a reminder why an infinite union is interesting in a measuring process.
Awesome videos
Welches Programm nutzt du?
Which programme he does use?
I have no idea what I am doing. I just graduated. But does these sets have something to do with that set paradox letter?
Many thanks for these good and helpful mini-lectures. And I would like you to direct me to a useful textbook on this theory from your point of view.
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.
Thank you.
Thanks so much
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 :)
Thank you, you are the best
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.
Beni buraya kadar getiren eğitim sistemimize teşekkür ediyorum.
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!