It is important to notice that the pointwise supremum of nonnegative measurable functions is also a nonnegative measurable function and this allows you to integrate it.
At 1:09:43, (ii) lim \int |fn-f| = 0 , not only finite. (corrected) At 1:11:11 it can be R bar, with infinity since f,g are integrable, and therefore \int f is finite by definition.
35:06 the measure of the pre image of z? Shouldn't it be the measure of the pre image of [z, infinity) ? Anyway, he is easily the best math teacher I've ever seen in my math learning journey.
@@jwp4016 Its still false, on preim[0,z) f will be < z... you need preim[z,oo) so that on that set f >= z and you will get a lower bound of the integral of f...
Dear Dr. i glads to follow such interesting lecture ...really really it was nice lecture. before this lecture i am not familiar with this concept after following your lecture things was clear for me thank you very much.keep it up
But the borel algebra contains a lot more than just open sets. You could write {s_i} as a countable intersection of the sets (s_i - 1/n, s_i + 1/n), each of which is open, and hence measurable. So the intersection {s_i} is also measurable.
there is an easier way to see this: the complement of a singleton is open in the standard topology. but the sigma-algebra is closed under complement-taking, so the singleton is in it. in fact, the borel sigma-algebra contains all the open AND closed sets, furthermore all the half-open intervals and so on
I was the one who could be easily mislead by this. I haven’t taken measure theory formally, but need to learn more for my upcoming program in finance! You’re series is definitely unambiguously presenting the math, and I won’t say that about anything else I’ve come across. Any book recommendations?
"A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson, it is a really nice and friendly introduction to measure theory. Has lots of intuitive explanations and includes illustrations, all the while being completely rigorous.
Please doctor, l need to solve this equations. : 1: show that the step function f (X) =[x] is measurable function on [1,3] And 2: if f is measurable function over a measurable set E then f is measurable over any measurable subset A of E
That was maybe the best explanation I have ever heard of the quotient space. Brilliant work!
It is important to notice that the pointwise supremum of nonnegative measurable functions is also a nonnegative measurable function and this allows you to integrate it.
Greatest mathphysics on line, and loved the student interaction.
Such a great lecture. Absolute clarity in all of the explanations
I tried to self study and different videos, your explanation is sooo clear and easy to understand, wish I could attend your class!!!!!
Wonderful teachig style with sufficient content covered less than 2 hours. Thank you sir.
Hölders and Cauchys inequality should both be multiplications and not additions.
yes
The prof. meant Minkowski inequality
The important thing now is that I learned how to pronounce ö as in Hölder!
@@benpovar3914 no, he must have made a mistake. The Cauchy-Schwarz inequality is definitely with multiplication, and the Hölder inequality too.
@@benpovar3914 Minkowski is slightly different. ||f+g||
Great lecture. Much appreciated.
The clarity of Schuller's explanation is amazing.
Thanks for the lecture! You make Lebesgue integration seem easy.:)
Such a clear lecturer. Makes the difficult easy. Many thanks.
I like his wavy norm sign
It is clear and tidy.
At 1:09:43, (ii) lim \int |fn-f| = 0 , not only finite. (corrected)
At 1:11:11 it can be R bar, with infinity since f,g are integrable, and therefore \int f is finite by definition.
This really is a great video, I'll definitely watch more of your lectures! Helped me a lot, thanks.
Just a remark teacher chaucy inequality with product not a sum : ||
a great lecture
35:06 the measure of the pre image of z? Shouldn't it be the measure of the pre image of [z, infinity) ?
Anyway, he is easily the best math teacher I've ever seen in my math learning journey.
I missed 1:13:36
@@jwp4016 Why is it not just preim(z)?
Yes ... in the sketch he did preim(z) has zero measure
@@jwp4016 Its still false, on preim[0,z) f will be < z... you need preim[z,oo) so that on that set f >= z and you will get a lower bound of the integral of f...
lol he finally corrects that afterwards :P
this is an excellent lecture sir.
Dear Dr. i glads to follow such interesting lecture ...really really it was nice lecture. before this lecture i am not familiar with this concept after following your lecture things was clear for me thank you very much.keep it up
Amazing lectures. Is there a textbook(s) for this course?
Very clear. The set of your mistakes has measure zero.
QM lectures turn out to be great for anyone interested to start functional analysis
1:20:50 Progress to the direction of Hilbert and Banach spaces... and why this matters when one does solid state physics!
i remember these terminologies such as borel sets, sigma-algebras...etc.
thanks
25:03, 42:35, 1:12:28 (35:58), 1:25:17, 1:32:00, 1:35:10, 1:38:38, 1:44:03, 1:48:48
Why is the singleton {s_i} measurable, since a singleton in the standard topology is not open and therefore also not an element of the sigma-algebra?
But the borel algebra contains a lot more than just open sets. You could write {s_i} as a countable intersection of the sets (s_i - 1/n, s_i + 1/n), each of which is open, and hence measurable. So the intersection {s_i} is also measurable.
there is an easier way to see this: the complement of a singleton is open in the standard topology. but the sigma-algebra is closed under complement-taking, so the singleton is in it. in fact, the borel sigma-algebra contains all the open AND closed sets, furthermore all the half-open intervals and so on
Excellent lecture, Thanks.
Very understandable
why these lectures, youtube don't have automatical translation?
I was the one who could be easily mislead by this. I haven’t taken measure theory formally, but need to learn more for my upcoming program in finance! You’re series is definitely unambiguously presenting the math, and I won’t say that about anything else I’ve come across. Any book recommendations?
Real Analysis by Royden
Needs more examples
"A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson, it is a really nice and friendly introduction to measure theory. Has lots of intuitive explanations and includes illustrations, all the while being completely rigorous.
Good🤓
Are there any problem sheets for people viewing these lectures on You-Tube, are the Experimental lectures on You-Tube?
height times width (not with :D)
good joke 1:11:48
Please doctor, l need to solve this equations. :
1: show that the step function f (X) =[x] is measurable function on [1,3]
And
2: if f is measurable function over a measurable set E then f is measurable over any measurable subset A of E
very fine but where are proof of theorems
1:05 point aus... Oops, wrong language :-)
cocaine? :D
the gender jabs never get old