A horizontal integral?! Introduction to Lebesgue Integration

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[elidle]

As the King of Calculus, Riemann, faced the Dirichlet function, Riemann asked the function, “Are you Riemann integrable because you are continuous? Or are you continuous because you are Riemann integrable?” (note: this iff statement is not actually true) Riemann began to open his domain *”Malevolent Shrine”* and *cleaved* the area under the curve into slices approaching infinity. However, the Dirichlet function is continuous nowhere, and by *contradiction* the function simply stated “Stand proud. You're strong but Nah, I’d win.” Bernhard Riemann’s domain crumbled. In his dying moments, Riemann uttered the phrase “With this treasure, I summon…” Because in sets of measure zero always bet on *Lebesgue.* The proof was trivially concluded, and those who pioneered the techniques of calculus, the ones who formalised the integral, they would all bear witness to the one who is free. To the one who left it all behind, and his overwhelming *integrability!*

[canopusinthenorth]

So cool, more 3b1b inspired youtubers.. I am slightly jealous of kids today who are getting this level of education for almost free.

[TWI]

Clciked and thought it was 3b1b, glad I made that mistake, great video!

[Lucky]

This is excellent. Every time I've tried to understand lebesgue integration in the past it was presented as something complicated and hard to understand, but you've made it so simple. I'm sure there's more detail you didn't cover, since once video can only do so much, but you've actually given some good intuition what what a lebesgue integral actual IS, which is something I haven't been able to find in any of the many other explanations I've found. Thank you very much!

[tracyh]

At 7:19, E(x) should be the integral of x*P(x), not just P(x). The integral of P(x) is 1!

[yimoawanardo]

One thing I'll give as an advice : don't stop. I looked back at myself from after 6 years and I'm ashamed of how better I was at 14/15/16/17. You're at an awesome level. Make sure it's not your peak!

Awesome video :)

[drpkmath]

This is so cool for sure. Animated presentation is really good

[ryans]

Good video! You have an amazing talent of effectively illustrating complex ideas in a simple manner. Keep up the good work!!

[eliyasne]

7:25
Isn't it supposed to be the integral of
x p(x) dx

[BlueRaja]

The "limits of rectangles" explanation is what they teach you in high school calc, is how Newton/Leibniz thought of integration, and is correct to a good approximation, but it's not technically what a Riemann Integral is. Riemann Integrals (which are taught as a rigorous framework in Real Analysis class) are actually more powerful, and are able to integrate any function which is continuous "almost everywhere".

Take Tomae's Function, which is extremely similar to Dirichlet's Function that you showed:
f(x) = 0 if x is irrational, 1/n if x = m/n (rational)
This function has infinite discontinuities. However, it's discontinuous at every rational number and continuous at every irrational number(!!). Since there are only countably-many discontinuities, it's continuous "almost everywhere" and is therefore Riemann Integrable!

In fact, it was discovered in the 50's that a small tweak to the definition of Riemann Integrals makes them strictly stronger than Lebesgue Integrals - that is, every function which can be integrated using Lebesgue's Method can also be integrated using these new "Generalized Riemann Integrals", but the converse is not true!

[rv]

The idea that somehow the Lebesgue integral is "done horizontally" is misleading.

The Lebesgue integral is defined _exactly_ like the Riemann integral, except that for the Lebesgue integral the simple functions are linear combinations of characteristic functions of _measurable_ sets rather than just intervals.
Then, once you've defined it, you prove that the integral is the area of the subgraph, you prove Fubini's theorem, and the "Cavalieri principle": all this allows you to rewrite the integral "horizontally" so to speak.
No need to directly define the Lebesgue integral "horizontally".

[sindbadthesailor]

Please do more of these vidoes! Good explanation. Would love some intuative measure theory / functional analysis.

[MClilypad]

3:14 this is a bit too quick. While the probability for this one number is 0, we also have an infinite amount of other rational numbers it can be. So the total probability is an infinite sum of things. To determine that that is 0 you need some more arguments.

[erebology]

This video just inspired me to extend this result into a deep area of number theory!

[freddyflores]

This channel is gonna explode. Thanks for the explanation.

[alberttomasi]

Very clear and interesting explanation ! It was very cool to learn about the strengths of Lebesgue integral over Riemann integral. Thanks !

[MatesMike]

This channel has so nice quality! Congrats! :)

[sukkrivaavijayan]

Mate I have an lesbegue integration exam in a few hours and was nearly having a panic attack till this video came along, thanks so much! Subscribed!

[tastypie]

This channel deserves much more subscribers than it has now

[Lucky]

I really like the analogy to expected value. It's much more helpful than the coin analogy in Lebesgue's quote. The coin thing is too vague, but expected value is much more specific and so is a much more satisfying analogy.
I also really appreciate that you gave a specific real world example of when Lebesgue integration would be useful. Personally, I particularly like the example with circuits as I'm an Electrical Engineering major.