Lebesgue Integral Overview
Вставка
- Опубліковано 8 лют 2018
- In this video, I present an overview (without proofs) of the Lebesgue integral, which is a more general way of integrating a function. If you'd like to see proods of the statements, I recommend you look at fematika's channel, where he gives a more detailed look of the Lebesgue integral. In another video, I will give a specific example of a function for which I'll calculate the Lebesgue integral. Enjoy!
FINALLY SOMEONE WHO PRONOUNCES IT RIGHT
The French "es" before a consonant is an old version of "ê", like "crêpe" was "crespe"
Like for the English "hospital, hospitalization", we have "hôpital, hospitalisation" :)
So "Lebesgue" comes from "le besgue" (the old way of writing "the stutterer") which became "le bêgue" !
Bonne vidéo !
Wow, I knew that Lebesgue came from Le bêgue, but I didn’t know that bêgue had a meaning!
My favorite is fenêtre vs. défenestrer, it really shows that es becomes ê
"Easter" works like that too :
"Pâques", and the adjective meaning "paschal" is... "pascal".
Whoa, mind-blown!!!!!!! 😱😱😱
Well, in Hebrew it's לבג (pronounced lebeg) so no place for mistakes here :P
It appears to me as tho it happens all the time in maths in general, eg - L'Hospital's Rule...
paused on 2:21 and I must say: Dr Peyam, you give off so much positive energy that I'm instantly so encouraged to understand the topic well and I think that many lecturers should learn how to engage their students like you do.
Black Peyam Red Peyam!
What a great video in which Sir has given more than clear explanations . The best video on the subject I have ever watched !
lol that dude is always smiling lmfao
He's high on math.
Joker gas
well, sure he is. The man who understands maths, understands the world...
This guy is helping me get through quarantine
Very nice, writing my calc 3 exam tomorrow about measure theory and lebesgue integration. If you go deeper, you'll discover very powerful theorems like Fubini's or Transformation. I really advice everybody to study that field, it's awesome!
This comment is directed at ALL OTHER UA-camRS who have posted videos on measure theory and the Lesbesgue integral:
Dr. Peyam has just shown you above how to teach this concept. Every other youtube math teacher starts by rehashing how the Riemann integral works--usually showing off their prowess by integrating a complicated function--and then they go down a list, axiomatically, of Lesbesgue concepts, usually descending into the depths of PROVING THEM ONE BY ONE, thereby losing the attention of the viewer.
Then they close by verbally stating that the Lesbesgue integral simplifies and generalizes integration, but providing no graphic demonstration.
That's the abstract nonsense that normally passes for a pedagogical video on the Lesbesgue measure.
Thank you Dr. Peyam, for your pedagogy and your enthusiasm.
Thank you ❤️
Sou do Brasil, e encontrei os seus videos somente hoje. Embora tenho certa dificuldade em entender o idioma, gostei muito das suas aulas e da forma como explica. Parabéns Professor!!
11-12-2020. This is masterful and excellent! Great work.
Wow, this is the most mind blowing video I've seen of yours
so nice, you just saved one student struggling with real analysis
thanks..
Mind = blown !! I just finished solving a lot of problems in my introductory real analysis text. Can't wait to get into measure theory.
I have watched several of your videos, I think you come across incredibly well and clear!
Thanks so much!!
Great explanation, love your passion for math!
thank you Dr . I was happy with your lecture on the topic keep it up.
Always good maths made by a even better mathematician
Very clear explanation,thanks
Thanks for this vid , its the first time i even heard of the Lebesgue integral, gud stuff.
Wow this white board and the writings is crystal clear
Thanks once again, Dr Peyam! A couple of requests: Could you please list videos referred to (links would be nice!), and also give some indication of sequence / dependency of yr videos. I know this may change as you insert stuff between topics covered, but I'm sure yo cna come up with a system that will help us all! As usual, very clear. Sahzoom til we meet again!
Looking forward to see more and more videos of lebesgue measure.
You are the best teacher I have seen and I love you
❤️
Great course ! Thank you I realy enjoyed
Fematika is how you spell the channel, it's a good one for very high level pure math concepts
Excellent overview.
that was reallly useful, thank you very much
Very powerful theorem indeed! Nice video...
I'd love a video on measures of sets.
Fematika has a great series about that, check it out!
Amazing and mind blowing sir
Great teaching technique
Very very clear！Thanks a lot！
The first rule of non-measurable functions is you don't talk about non-measurable functions.
So true 😂😂😂
Brilliant. Thank you.
Totally MVP! Thanks Dr!
what a great touterial. i was wondered if you could make others on the measures
Thank you for teaching with smile like this :))
Looking forward to some examples.
Coming on Monday!
Came here from fematika. Feel like I should've done it the other way around haha
but seriously you both have epic channels
Thank you. I still enjoy the chalk and talk as well. Please do not leave it orphaned in the world.
Something where the Lebesgue integral is easier than the Riemann integral? I will stay tuned.
Looking forward to seeing a function where the Lebesque integral is easier than the Riemann integral.BTW: the whiteboard is much easier readily (as many others have stated already)
First, some functions are Lebesgue integrable but not Riemann integrable, such as the indicator function of the rationals of an interval. Then, any Riemann integrable function is also Lebesgue integrable and both integrals are equal. This allows the use of the dominated convergence theorem when all functions are Riemann integrable, which is much easier and more general than any convergence theorem for the Riemann integral: no need to worry about uniform convergence for exemple.
Just look at Chebyshev’s Inequality proof with Riemann vs Lebesgue. Lebesgue is great for proofs and such. But it's rarely computable (or at least easily and generally so).
He reminds me of my functional analysis professor....
holy cow man this is what i was looking for.
Instasubscribed
THANK YOU
Oh sweet, this is the classical example.
Never clicked a video so fast in my life xD
I love you, this video is great :DD
The baby way to do an integral :)
Fematika shout out!
and I dont know if you said in the video so: part where you talked about countably additive function, it is related to "vitali hahn saks theorem"
So I just realized (retrospectively), that this function consists of line segments of the devils staircase where each of those line segments is bisected by y=x. If you look at any segment alone, and y=x, you get two symmetric triangles formed by the segment line and y=x (and vertically by the boundary of the segment). One of those triangles is below the line (and thus part of the area, but above y=x), and one is above the line (thus not in the integral, but of the same size as its counterpart, allowing you to make a trapezoid by "flipping" the left triangle (above y=x) to the right side (below y=x), preserving area). If you did this to every segment from the staircase, you would end up with all those trapezoids standing side-by-side to make a triangle of area 1/2. Of course, this would not involve lebesgue interals though. Nonetheless, this example is appreciated since it clearly explains how to use lebesgue integrals.
Hello sir , I am unable to get the lebuguese integrable function on [0,1]. Which one of the below is correct sir .
-->X.ln(X)/(1+x²)
-->sin(pi.x)/ln(X)
-->ln(X)(ln(1-x))
-->ln(X)/root(1-x²)
Thank you Dr Preyam for your magnificent video lecture, If you don't mind I have a silly question: In 4:27 you wrote that the measure of A is the integral over R of f(x), why did we integrate over R (the whole real line) instead of integrating on the open set A (i.e. (2,5) as you showed in your example)? Does R here mean the whole real line or a subset of R in which A is "defined." Thank you in advance and have a wonderful day.
The whole real line
@@drpeyam Thank you very much for the quick answer.
Sir @16:17, can I approximate the function using Taylor series?
The convergence of f in the step 2 is puntual?
Love your videos, discovered you not so long ago but it's awesome and now even better with the White board...
PS : lefties will rule the world :p (love from a French left-handed boy :D)
Great!!
the visualization of dominated convergence theorem was top knotch.
Thank you!!
let f(x)=sinx /x when x>0 and 0 when x=0.....is it riemann or lebesgue integrable? WHYYY? please reply me i cannot solve this.
In the dominant convergence theorem should it be pointwise convergence almost everywhere?
I think so
Thsi board is readable -.. good for you
Does the lebesque integral only output scalars? Is there an indefinite lebesque integral? I'm still very new to Lebesque. My honors Calc professor was a set theorist and just mentioned lebesque one time whilst defining Darboux's integral.
There’s no indefinite Lebesgue integral that I know of, but I think in the same way that I did here, you can make Lebesgue integrals spit out anything that’s in a Banach space (which could be real numbers, complex numbers, even functions), but in general the classical Lebesgue integral of a function spits out only a number.
Dr. Peyam's Show Good to know! Thanks, Dr. Peyam, and bravo on making another enlightening, inspiring video! I look forward to the next.
@13.30 - it's over 9000!
Sir please halp me 🙏🙏
Q let f:[0 1] →R be given by f(x)=0 if x is rational and if x is irrational then f(x)=9^n
n is a number of zeroes immadiatlay after the decimal point in the decimal representation of x then the lebesgue integral
0 to 1
∫f(x) dx
how about int[0,1/pi] |sin(1/x)| dx? does it lebesgue integrable? and does it converge? i mean intuitively the area is less than the area of a rectangular does it mean that it is converge?
That’s an interesting question! I might be wrong, but sin(x)/x from 1 to infinity is not Lebesgue integrable (but improper Riemann integrable), so I’m guessing no?
@@drpeyam when x goes to 0⁺, does the |sin(1/x)| relate to cauchy sequence? because this reminds me of cauchy sequence, you said that when people grouping around doesnt mean that there is something
@13:38 - was that a DBZ Over 9000 reference?
Can you prove Fubini's Theorem ?
Could you please do a video on the Henstock-Kurzweil integral? I find it confusing.
I don’t know what that is
@@drpeyam Its a generalization of the Rieman integral; there's a Wikipedia article about it.
4:32 So what does it mean to be able to "count" something which is infinite?
A set is countable if there is a way to match the natural numbers with the set's objects ('1-1' match) You can have infinite countable sets (Q) or non countable ones (R)
Why do you say WLOG the Ai are pairwise disjoint? I don’t see the need for a WLOG there, other than than great video bro! I also have some math videos , mostly basic number theory
Case for most general function is missing. Show how to Lesbegue a sin function. Is this not more complicated then Riemann integral?
I'm not even a mathematician and my mind is getting blown.
Fun fact: all Riemann integrable functions are always Lebesgue integrable, but this is not the same the other way around.
Peyam for life.
What if f+ and f- are infinite? Does the integral of f exist in this case?
No, it would be undefined
What is the name of the channel referenced at 0:38?
Fematika
@@drpeyam Thanks! x 10^6
Awesome :3 excuse me... I have to watch some examples c:
Por favor add+ legendas em português.
I was taught that the function with 0 on Q and 1 on I or viceversa is not integrable because the discontinuities set is the whole domain, which doesn't have measure zero. By Lebesgue theorem of integration, the function can't be integrated
No, it’s not Riemann integrable, but it is Lebesgue integrable!
@@drpeyam That was totally my fault. I misunderstood my 'Lebesgue theorem' which is actually referring to Riemann integrals, not Lebesgue integrals, which I haven't even been taught about in class. Thanks for help.
how can you write script Q that good?
only 3 more video on measure?! wow, this is kind of amazing considering how complicated it can get... with σ-algebra and such... actually, it would be great if you can make videos about measure!
p.s., this is the first time i see the function of 1_A written with '1', i always so it with the letter 'chi'(youtube does not support the script of this letter).
p.s.(s). if you decide that infinity measure=not integrable then saying "
Hahaha, it comes from 15 years of experience of writing Q’s! And fematika has a great series on measures, you should check it out! And I’ve seen both notations used :) Not sure I understand your last question, though!
Dr. Peyam's Show I'll watch his series, I like to see how other teach(because I'm horrible at it, I'm trying to study how to do it).
You can ignore the last part, it is a matter of how one define infinity, not something that cause harm
Yuval Paz in measure theory we normally use the extended real line, so that '< (infinty)' does make sense
Shakespeare said, 'to be or not to be' 😀
But he forgot to add: almost everywhere.
Riemann can't integrate it? No problem. Dr. Peyam has the problem...
😎
In Lebesgue.
rajesh kooterpali
Virgin Riemann Integral vs. Chad Lebesgue Integral
And now my head hurts :(
∫ f = lim ∫ fn
Is that lebesgue convergence theorem?
Well, it depends :P If fn are simple, then it’s just the definition. But in general it depends on the assumptions; If fn is increasing with respect to n then it’s the monotone convergence theorem; if the fn are dominated by g then it’s the Lebesgue dominated convergence theorem
Payam? Baba hamshahri !!
TUCK PROPERLY BRO
😆😆😆😄😄😄😄😀😀whaaast is thaaa difference bethween Rhemaasn inthhegral n Leeeeesbegue ? Khaan u doiooh thaaat? Eiha uum akhble to unthherstand whaaasts yooour khooooncept ishhhhh😃😃😄😀😀
11:49 da wae
could you perhaps try making your videos a bit shorter? baba jan kheyli toolanieh
at the end i feel nervous not educated.
One does not speak of non measurable functions. LOL
Hahahaha
@@drpeyam Thank you very much for the explanation
Bro are you kaka doing math