It's funny how this video in particular among a the other ones in the playlist has a lot of views. I guess it's because there really are not many (if at all) good introductions to measure theory available. I mean, this lecture was very illuminating for me, since I never really unterstood what the Lebesgue integral was.
Hi Jackozee. Take a look at Daniel Cohn's "Measure theory". It helped me very much to grasp measure theory and Lebesgue measure. Another very good book is Heinz Bauer's "Measure and Integration Theory".
I sometimes ask myself what is a measurable set. He said 57:00 What is a measurable set? Don't ask it, you should ask what is mesurable space? and the elements are called mesurable sets.
Ja-Keoung Koo it's interesting. I would say it's just a name given to the constituents of our structure. In the structure of a vector space, the constituents a vector s, but what really are vectors? The point is, don't think from the ground-up but think top-down.
In standard mathematics you don't define the Lebesgue measurable sets as only the Borel sets of R^n because otherwise you can have sets that are subsets of 0-sets that are not measurable. Standard is that you use the completion of Borel sets.
two years late but... I think he is posing it as an unproven theorem that the lebesgue measure is the only measure that can satisfy the property given. Im unsure if this is true since he did not give the proof.
Thanks professor! Really enjoyed yet another great lecture! Just a small and humble notice: In the context of "almost everywhere / almost surely": The subset A of M for which the statement does NOT hold is not necessarily measurable itself. Yet this subset A has to be subset of a null set (event), i.e., of a measurable subset of M with zero measure (probability). Also, I often find a finite measure space being defined as one that (simply) has \mu(X) < \infty. Your definition seems to refer to a \sigma-finite measure (space). I'm far from an expert, and probably wrong, but maybe someone can shed a light on this in the comments.
You are right regarding null sets. People often confuse null sets with sets of measure zero: a null set is a (not necessarily measurable) set that is *contained* in a measurable set of measure zero. See e.g. Exercise 4.12 in René Schilling's "Measures, Integrals, and Martingales" (a book which I can highly recommend). You are also right regarding the definition of sigma-finite measures as opposed to finite measures, see e.g. Definition 4.2 in Schilling's book.
Just a small word of warning: the definition of a sigma-algebra in this video (at 7:58) is not correct. The common definition (see, e.g., en.wikipedia.org/wiki/Sigma-algebra#Definition) omits the disjointness assumption in point (iii). Other equivalent definitions exist, however, the definition in this video is really wrong as it would render the set { {\emptyset}, {1,2}, {3,4}, {2,3}, {1,4}, {1,2,3,4}} a sigma-algebra. Note that this set is not closed under intersections.
You are right that this definition is wrong. There is often some mix-up with the disjointedness because you demand it for the definition of a measure later. Therefore, it is always good to consider some examples. Then you easily find such oversights.
Like all researchers in Mathematical Physics, Prof Schuller is more a "geometer" than a pure mathematician. From the start of the lecture you can feel he is not completly confortable with this subject. Mind you, he is a very good teacher and these are very difficult subjects to teach (and to understand...even for mathematicians). Furthermore, he is not using notes... (which does not mean he does not prepare the classes : he does)... I think I know why he does that: for the same reason I do, to challenge himself to present the lesson more naturally, intuitively. But this is not an easy exercise, even for someone as talented as he is. His classes on GR are much better
@28:49 I believe there is possibly a slight mistake, which I am mentioning just to make this absolutely awesome lecture more follow-able: "continuity from above". Shouldn't the intersection of n decreasing sets A sub n, where A sub 1 contains A sub 2, and A sub 2 contains A sub 3, et cetera till A sub n be equal to the smallest subset in this Russian doll, i.e. A sub n, as opposed to A, which was defined earlier in the lecture as the union of all the subsets?
Just for anyone interested. Your conclusion is true for a finite sequence {A_j}_j∈J, however for a countable set {A_j}_j∈ℕ the smallest subset (by notation) is A=∩_j∈ℕ A_j. Notice that A is NOT part of the sequence, but identifies its smallest member.
Axel Mothe for the second claim I guess we could use the fact that for a monotonic function the set of discontinuity is countable and for each of the "continuous portions" we use what he said. Am I right?
Theorem that states the composition of two measurable functions is measurable is incorrect. Consider f to be a simple function on a set with measure zero, g to be Cantor function, then the pre-image of a measure zero set under Cantor function can be a non-measurable set contained in the complement of Cantor set.
He's trying to show that all half-closed intervals [a, b) are measurable. He's doing this partially to disabuse students of the notion that it is only the open sets which are in the Borel sigma algebra, which they might believe since they used the set of open sets as the generating set for said sigma algebra. An easier way to do this would be to look at the intervals (a, c), (b, d) where a < b < c < d. Then (R \ (a, c)) ∩ (b, d) = [c, d) which by the axioms of sigma algebras is also in the sigma algebra. Though simpler, this proof doesn't make use of the fact that countable intersections of measurable sets are measurable, which is probably also something he wanted to remind the students of.
i think that he somehow forgot to mention that during continuity from above super set A1 represents A....not the intersection of that set ..intersection leads into lim n tends to infinity An.... so the measure upto infinity leads to measure of A1..... idk....
That's not true, the intersection of such decreasing sets leads to the smallest set (denoted by A) in the countable sequence {A_j}_j∈ℕ. A_1 instead represents the largest set in the sequence.
Yep. Now try to construct, mathematically, the Banach-Tarski partitions. That being said, I am a mathematician, so listening to Vsauce usually makes me jump erraticaly on my chair for all the stupid and false stuff they publish. They give the illusion of knowledge, which I find worse than simple ignorance.
@@sardanapale2302 Disagree, the vsauce video was not one of these stupid zero knowledge video (for the regular folks not the mathematicians or physicists). It is on the maximum level of complexity and formalism. That is how "pop-science" should look like. It never conveyed that after that video you can prove or used or really understand the BT theorem.
I must say I still prefer the idea of a “reasonably behaved function”, as fuzzy as it is, over the endless cryptic technical definition salad any time.
The problem is that then your theories become ill-defined and at the end you don't understand anything you write down. Maths can seem difficult but at the end it is rigorous and it will give deep understanding of the material
Reimannla lebgela bill gatesla cs wado premanla oya level eka mata wadi uni awata🙂 Apita ba mam yanwa Krana giyoth wene hirwena eka A/l ugan chun eke enwa Mewa wadi.
A purely theoretical quantum mechanics course is typically taught in semester 3-5 in Germany, most often the fourth. This lecture series here is way more mathematical than usually, but my guess is that the target audience is still about that semester, while some more advanced (and some curious younger) students may be attending, too.
![[LIVE] : ONE ลุมพินี 90 | คู่เอก "ก้องไกล vs แอนตาร์"](http://i.ytimg.com/vi/VugIvOstzRI/mqdefault.jpg)
This guy clearly knows his stuff. He has the theory so well under controle that he was able to realise quantum telportation at 26:51
He actually lives on a torus, no quantum teleportation needed.
Physicists are the scourge of science.
Hilarious
God have mercy on the poor souls that are destined to live on planet earth without ever coming in contact with these amazing lectures
This lecture stands on its own! Great intro to probability theory as well!The sigma algebra stuff is crucial!
I wish this guy uploaded more of his lecture videos.
It is important! hence it is defined that way.
He is the guy that I really learn push forward and pullback from. The teaching of this professor is natural and I like it so much
I loved this, first thorough introduction to measure theory I've ever had
It's funny how this video in particular among a the other ones in the playlist has a lot of views. I guess it's because there really are not many (if at all) good introductions to measure theory available.
I mean, this lecture was very illuminating for me, since I never really unterstood what the Lebesgue integral was.
Hi Jackozee. Take a look at Daniel Cohn's "Measure theory". It helped me very much to grasp measure theory and Lebesgue measure. Another very good book is Heinz Bauer's "Measure and Integration Theory".
You are absolutely right
There's one from "The bright side of Mathematics"
Check out, I think you'll like it.
@@carlesv1488Lol you mean Donald L. Cohn's "Measure Theory"
I sometimes ask myself what is a measurable set. He said
57:00 What is a measurable set? Don't ask it, you should ask what is mesurable space? and the elements are called mesurable sets.
Ja-Keoung Koo it's interesting. I would say it's just a name given to the constituents of our structure. In the structure of a vector space, the constituents a vector s, but what really are vectors? The point is, don't think from the ground-up but think top-down.
These videos helped me complete my PhD for math. Thank you so much
Sublimely insightful lecture. I understood everything after the professor presented it. He commands respect.
The prof is a treasure. He knows the subjects in a far greater depth than the one he chooses to teach.
In standard mathematics you don't define the Lebesgue measurable sets as only the Borel sets of R^n because otherwise you can have sets that are subsets of 0-sets that are not measurable. Standard is that you use the completion of Borel sets.
two years late but... I think he is posing it as an unproven theorem that the lebesgue measure is the only measure that can satisfy the property given. Im unsure if this is true since he did not give the proof.
Thanks professor! Really enjoyed yet another great lecture!
Just a small and humble notice: In the context of "almost everywhere / almost surely": The subset A of M for which the statement does NOT hold is not necessarily measurable itself. Yet this subset A has to be subset of a null set (event), i.e., of a measurable subset of M with zero measure (probability).
Also, I often find a finite measure space being defined as one that (simply) has \mu(X) < \infty. Your definition seems to refer to a \sigma-finite measure (space).
I'm far from an expert, and probably wrong, but maybe someone can shed a light on this in the comments.
You are right regarding null sets. People often confuse null sets with sets of measure zero: a null set is a (not necessarily measurable) set that is *contained* in a measurable set of measure zero. See e.g. Exercise 4.12 in René Schilling's "Measures, Integrals, and Martingales" (a book which I can highly recommend).
You are also right regarding the definition of sigma-finite measures as opposed to finite measures, see e.g. Definition 4.2 in Schilling's book.
Just a small word of warning: the definition of a sigma-algebra in this video (at 7:58) is not correct. The common definition (see, e.g., en.wikipedia.org/wiki/Sigma-algebra#Definition) omits the disjointness assumption in point (iii). Other equivalent definitions exist, however, the definition in this video is really wrong as it would render the set
{ {\emptyset}, {1,2}, {3,4}, {2,3}, {1,4}, {1,2,3,4}}
a sigma-algebra. Note that this set is not closed under intersections.
I should have added that otherwise I think Dr. Schuller is doing a great job!
You are right that this definition is wrong. There is often some mix-up with the disjointedness because you demand it for the definition of a measure later. Therefore, it is always good to consider some examples. Then you easily find such oversights.
This is fixed later in the video.
Like all researchers in Mathematical Physics, Prof Schuller is more a "geometer" than a pure mathematician. From the start of the lecture you can feel he is not completly confortable with this subject.
Mind you, he is a very good teacher and these are very difficult subjects to teach (and to understand...even for mathematicians).
Furthermore, he is not using notes... (which does not mean he does not prepare the classes : he does)... I think I know why he does that: for the same reason I do, to challenge himself to present the lesson more naturally, intuitively. But this is not an easy exercise, even for someone as talented as he is.
His classes on GR are much better
This guy is one of the best lecturers I have ever seen
The best explanation of Borel sigma-algebra ever !! Suggest some Measure theory Book Mr. Fredric!
R.G Bartle the elements of integration
Dear professor why do u teach so well. I envy how lucky ur students are to have u as instructor
The versatility of the Prof is astounding.
1:03:00 Memo: all you need to know for the Lebesgue measure you already learned in kindergarten .
21:42, 34:58, 37:18, 40:18, 44:48, 49:45, 56:36, 1:04:27, 1:14:09, 1:17:12, 1:32:17, 1:39:13
What a WONDERFUL professor! Thanks.
The best explanation of Push-forward measure too!!
excellent review ... thank you so much professor
@28:49 I believe there is possibly a slight mistake, which I am mentioning just to make this absolutely awesome lecture more follow-able: "continuity from above". Shouldn't the intersection of n decreasing sets A sub n, where A sub 1 contains A sub 2, and A sub 2 contains A sub 3, et cetera till A sub n be equal to the smallest subset in this Russian doll, i.e. A sub n, as opposed to A, which was defined earlier in the lecture as the union of all the subsets?
Just for anyone interested. Your conclusion is true for a finite sequence {A_j}_j∈J, however for a countable set {A_j}_j∈ℕ the smallest subset (by notation) is A=∩_j∈ℕ A_j. Notice that A is NOT part of the sequence, but identifies its smallest member.
Hello I have one question, At 1:21:00 is this not true only for continue function (not monoton function) ? Thanks you
Axel Mothe for the second claim I guess we could use the fact that for a monotonic function the set of discontinuity is countable and for each of the "continuous portions" we use what he said. Am I right?
Can you Upload more lectures ?
Brilliant, anyone happens to have the problem sheet? I'm really interested.
Dr Schuller is SO PISSED by people coming late and making noises during the class he is about to explode everytime ahah
2 things i learned in this video the professor hates late arrivals to his class and he hates talkers in class
yeah who doesnt lol...just stay home dont come
Theorem that states the composition of two measurable functions is measurable is incorrect. Consider f to be a simple function on a set with measure zero, g to be Cantor function, then the pre-image of a measure zero set under Cantor function can be a non-measurable set contained in the complement of Cantor set.
Cantor function probably won't appear in physics
@@theodoreree7100 : wrong. Example "tesselations" in "quantum gravity". More subtly, as spectra of certain operators.
What you mean are Lebesgue measurable functions? It is not the same as the measurable map stated at 1:13:48.
this is good stuff, anyone got a link to the lecture notes?
Theres a write up linked on reddit somewhere iirc
Is there any way to get the excercise sheets for these lectures?
Very nice presentation. Thank you very much.
I don't understand what he's trying to prove as from 55:00 - 58
He's trying to show that all half-closed intervals [a, b) are measurable. He's doing this partially to disabuse students of the notion that it is only the open sets which are in the Borel sigma algebra, which they might believe since they used the set of open sets as the generating set for said sigma algebra.
An easier way to do this would be to look at the intervals (a, c), (b, d) where a < b < c < d. Then (R \ (a, c)) ∩ (b, d) = [c, d) which by the axioms of sigma algebras is also in the sigma algebra. Though simpler, this proof doesn't make use of the fact that countable intersections of measurable sets are measurable, which is probably also something he wanted to remind the students of.
Wow I understood everything
Thank you
Excellent presentation Sir.
You are great Professor! I love you!
What is the difference between a sigma-algebra and a topology?
excellent explanation, well behaved class
Nice anecdote on corollary!
Hello, excuse me profesor, but what is the model of the camera, you use to record the class?
Very cool doctor, I like the this section of math.
2:40 was intense man
I didn't know what's gonna happen
They were already using this stuff way back in the day to build the radar fire control systems to aim the big 88.
i think that he somehow forgot to mention that during continuity from above super set A1 represents A....not the intersection of that set ..intersection leads into lim n tends to infinity An.... so the measure upto infinity leads to measure of A1..... idk....
That's not true, the intersection of such decreasing sets leads to the smallest set (denoted by A) in the countable sequence {A_j}_j∈ℕ. A_1 instead represents the largest set in the sequence.
Brilliant stuff!
Would it be possible to access to the problem classes and the problem sheets? Please!
@Frederic Schuller, Maybe input/release legend?
I don't understand what the first instructive example is about
Thank you
Vsauce has a great video on Banach-Tarski paradox!
Yep. Now try to construct, mathematically, the Banach-Tarski partitions.
That being said, I am a mathematician, so listening to Vsauce usually makes me jump erraticaly on my chair for all the stupid and false stuff they publish. They give the illusion of knowledge, which I find worse than simple ignorance.
@@sardanapale2302 Disagree, the vsauce video was not one of these stupid zero knowledge video (for the regular folks not the mathematicians or physicists). It is on the maximum level of complexity and formalism. That is how "pop-science" should look like. It never conveyed that after that video you can prove or used or really understand the BT theorem.
@@sardanapale2302 smh take your elitism elsewhere
Thank you!!!
This guy is brilliant!
.
I must say I still prefer the idea of a “reasonably behaved function”, as fuzzy as it is, over the endless cryptic technical definition salad any time.
The problem is that then your theories become ill-defined and at the end you don't understand anything you write down. Maths can seem difficult but at the end it is rigorous and it will give deep understanding of the material
Can someone send me the link to lecture 01 of measure theory?
Reimannla lebgela bill gatesla cs wado premanla oya level eka mata wadi uni awata🙂
Apita ba mam yanwa
Krana giyoth wene hirwena eka
A/l ugan chun eke enwa
Mewa wadi.
Maths sp ghila me amru subjet egemagan hida lami wage🙂
Mamam try kal ba amrui mama yanwa.
i took measure theory at university of houston in 2007, with professor auchmuty
May I ask in which semester the students are who are attending this lecture?
A purely theoretical quantum mechanics course is typically taught in semester 3-5 in Germany, most often the fourth. This lecture series here is way more mathematical than usually, but my guess is that the target audience is still about that semester, while some more advanced (and some curious younger) students may be attending, too.
Awesome!
I like this lecture! Thank you
thank you ++
1:33:30 nice pun.
Nice!!!!
He throws statement like punch. But he is subject expert.
Aah, what is the measure of man.
That FKR had to sit right there. What a tool.
Unmeasurable intelligence quotient
кайф
Lmao I hate to be the student in his class but great lecturer, questionable attitude 😅🤣
What do you mean with questionable attitude?