3 Integrals You Won't See in Calculus (And the 2 You Will) 

Bear witness to his overwhelming integrability!

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Dr Sean from iCarly!

1 ☑️ 2 ☑️ 3 what 😱

Just a small correction. Ito integrals are martingales under the additional (square integrability) condition, but in general, stochastic integrals with respect to martingales (e.g. Brownian motion) are only local martingales rather than true martingales. But like you said, it is a bit technical.

Thanks so much! I'm glad you liked it

And I am astounded how arbitrarily complicated it can be presented by some teachers. 😋

A probability of 1 means quite literally every time

@@DistortedV12 not necessarily, if you have infinite options. if you choose a random integer, the probability of not getting 7 is 1, but you can still get 7

nah, i don't think that's right.

Right! Thanks for your comment. I was thinking of sampling X_1, X_2, X_3, ... and meant that we'll see an irrational number each of these times, even if we keep sampling forever (countably infinitely many times). The union of countably many probability 0 events still has probability 0. But if we have uncountably many uniform random variables, then we may indeed get some rational numbers! We'll get irrational numbers almost every time.

@@DrSeanGroathouse Unfortunately I still think that's an incorrect interpretation of the probability here. Even if you only sample one point, it's still possible to get a rational number, for the simple reason that rational elements exist in the set [0, 1]. It would have probability 0, like you say, but probability 0 doesn't equate with impossibility. Another way to think about it: we agree that the probability of choosing any individual number from [0, 1] is 0. If probability 0 implied impossible, then this would mean that it would be impossible to select any element in the set!

Thanks for sharing 😁

Oye tío, ¿Tortilla de patatas con o sin cebolla?

Thanks so much!

So true.

Thanks! I'm glad you liked it

Thanks for the idea! I added it to my list for future videos

What isn't said in this video is that you can integrate any "measureable function" (ie a function for which the inverse image is "smooth" enough) with respect to any measure. Normal integrals like Riemann's are just integrals with respect to the lebesgue measure, but there are many more measures one can use. For example, if you integrate a function with respect to the counting measure, you get a series. Probability measures can also be used, and as a random variable is just a measurable function, you can integrate it with respect to the probability measure, and it gives you its expected value : E(X)=∫X(𝝎)dP(𝝎).

use a process that generates only rational numbers then add pi

@@gdclemo I guess rationals and irrationsals are of the same cardinality then

@@fightocondria Nope. You didn't say it should generate ALL the irrationals, but that it should only generate irrationals.

At least you are rational about it. 🤭

Newton and Leibniz got ideas about change and little bits down, but did not establish a formal rigorous foundation.

I image it comes from irrationals being uncountably infinite.

Probability being zero does not mean something is impossible in non discrete cases. Imagine you have a uniform distibution. If you imagine probability of one point be bigger than 0, lets say some x > 0, then, because disribution is uniform, you would have probability of any other point also be x, but there are infinitely many points, so total probability wont add up to 1. If you are interested check measure theory courses

The example in the video is the Dirichlet function, there are many videos that go into it. But the idea is that since in the real number line there are uncountably infinite numbers between 0 and 1, but only countable infinite rational numbers, the density of rational numbers between 0-1 is infinitely smaller (the so called measure μ), and they are overwhelmed by the amount of irrational numbers. The total Lebesque integral will be: 1*μ(rational) + 0*μ(irrational) = 1*0 + 0*1 = 0

@@dubvascl5840 Hmm okay. That's counterintuitive but I suppose I've got a reading list now. (Since I realize I didn't mention it before), I've only been up through a calc 2 course, does measure theory rely on much/anything beyond about that level?

@@TheTinyDiamond I dont know system of courses in US, but it should not. I my university i had Lebesgue integral at the end of 2 semester

The Lebesgue integral is constructed by extending a measure from the intervals to many other subsets of the real numbers (although not all, which is precisely why measure theory exists). This measure gives the interval (a,b) measure b-a. If you fix a point x, you can see that {x} is a subset of (x-epsilon,x+epsilon) for any epsilon larger than zero. But the measure of this interval is 2*epsilon. So the measure of {x} has to be less than 2* epsilon for any epsilon larger than zero - so it must be zero. The measure of a countably infinite set is just the sum of the measures of its points, which in this case is just zero. So the measure of the rationals is zero. But since there are uncountably many irrationals, you cannot apply the same logic.

Thanks for the idea! I added it to my list for future videos

it's also in any pure maths undergrad, in europe anyways

@@oke5403 I don't think pure math students were forced to learn it in my country (Brazil), but I know your pain

In Germany, the Lebesgue integral was first year stuff. From my perspective, the jump from Lebesgue to Ito was quite harsh. While the first three were all part of the first semester course for me, the last one I encountered just in my Phd studies.

@@patriziosommatinese1820 Damn, Lebesgue at the first year would be a little too much for me, at least depending on how rigorous was required. Itou integral I'll have to learn now in my masters degree because I'm working with a stochastic model

Here in Italy, it's second year stuff for mathematicians and, in less depth, for physicists.

that is inprope integral

@@leolrg2783 hummm. It doesn't matter. What I am saying is about the equivalence between Riemann and Darboux integrals. I am pretty sure this equivalence only works for functions that have sup in any interval of their domain, while Riemann integral will work for some functions that don't.

@@samueldeandrade8535 No. A function is rieman integrable then it must be bounded, so that sup exists.

@@leolrg2783 not true. Riemann integrable functions need to be bounded almost everywhere. For example, the function F(x) := 1/x, if x is rational, F(x) := 0, otherwise, is Riemann integrable, but NOT bounded.

@@leolrg2783 what you are saying is valid IF the function is continuous. If the function is NOT continuous, it may be valid or not.

Same with 1/3. Is it that these rational numbers are so greatly outnumbered that the possibility of picking one is negligible. Still, in precise math, I would still expect it to be non-zero. If someone could explain, I would be grateful.

@@michaelodekunle6591 The probability of an event being 0 does not mean the event is impossible. That's the "tl;dr" of it. An event can be possible and have probability 0 of occurring, as you've pointed out, when the sample space is infinite.

I think it has to do with the different countabilites, but not sure.

Itô for the masses? Blasphemy! 😆

They are always either French or German in mathematics, especially high level maths. LOL

​@@accaciagame1706 maybe 100 years ago. But in modern mathematics you have people like Nash, Turing, Conway and Feynman

@methatis3013 If we exclude New Zealand which has a tiny population, France still has the highest Fields medal per capita and the second net highest winners after the US. The French are unusually good mathematicians even to this day. Nash wasn't that good a mathematician. He just came across a very useful but simple Game theory concept. No where in the league of Newton, Poincarre, Euler etc.

@@methatis3013 nobody's saying English-speaking people are bad at maths. Only bad at even _trying_ to pronounce foreign names correctly. Not that the French are any better. Maybe it's a type of ignorance born of their colonial history, dunno.

Gee I wonder what 1/3 means

Realistically, there is uncertainty in everything. Exact values only exist as mathematical concepts. Tell me, do you think it's possible to cut a 2x4 to exactly 3.5 feet?

​@@InternetCrusader-rb7lsexactly, some numbers only have infinite decimals because of our number system,

A wild Axiom Of Choice appeared! 😄

😂

!La magia de este cine es *integral* ! 🤭