Lebesgue Integral Example

As promised, in this video I calculate an explicit example of a Lebesgue integral. As you'll see, it's a much more efficient way of calculating the area under that curve. Finally, I'll present a really cool way of doing this problem. Enjoy!

[meccamiles]

Your enthusiasm is contagious. I wish all professors were as inspired as you.

[joseantonioghz]

There is also a visual proof. I saw the function and realized that is bounded by 1 so the function lives under the square of legth 1, and also saw its symetry, if you turn it over you can't tell the difference, so it cuts the square in half hence the integral is 1/2.
Very good video!

[snakefang]

When I saw the staircase-looking graph, I went for a more geometric approach:
1) The function is self-similar, so the area from (0, 1) is six times the area from (0, 1/3).
2) The graph from (2/3, 1) is simply the graph from (0, 1/3) shifted up by 1/2, so the total area from (0, 1) is equal to twice the area from (0, 1/3) plus the rectangle from (1/3, 1), which has area 1/3.
3) Using these two facts, you can make a pretty simple system of equations: using A as the area from (0, 1) and B as the area from (0, 1/3), A = 2B + 1/3 and A=6B. Therefore 6B = 2B + 1/3 -> 4B = 1/3 -> B = 1/12 -> A = 1/2.

I always like geometric approaches when they're viable, and self-similarity is just fun to play with

[michalbotor]

i really enjoyed your example. it nicely showed how the lebesgue integration actually works and also shed a lot of light on the staircase function itself. i also liked your symmetry trick instantaneous calculation at the end.
enriched with the wisdom of the latter, now when i think about it, one can also correctly guess the area under the staircase function purely from the origami like reasoning as follows:
imagine that we have a flat, square sheet of paper of area 1. let us first fold it in half horizontally, then unfold it, then let us fold it in half vertically, then unfold it. these gives us back a flat, square sheet of paper but with two auxiliary creases: horizontal one and vertical one, that will soon come in handy. let us now fix n and draw in the direction of one of the diagonals of our sheet of paper the graph of the n-th simple function approximating the staircase function. let us next cut the sheet of paper along it effectively dividing the sheet of paper into two pieces: upper-left and lower-right. let us then take the lower-right piece and flip it first with respect to the line spanned by the horizontal crease and then flip with respect to the line spanned by the vertical crease. when we finish, the lower-right piece wil be covering exactly the upper-left piece, from which we will draw a conclusion that their areas must be the same. and since the must both add up to 1, they must be equal to 1/2. last but not least we will notice, that since this reasoning is true for every n, the same must be true also in the limit, i.e. in case of the graph of the actual staircase function.
i know its a mouthful in text, but that's because i tried my best to describe it as vividly and precisely as possible.
in practice, it's really instantaneous.

[diegoostoja-kowalski]

Who else figured out it's 1/2 when our hero showed the staircase picture?

[kurtu]

This is such an easy process to follow. So easy that the only step I didn't get was the evaluation of the convergent series (2/3)^k from k=1. I had to review them to figure out that you factored out 2/3 so you could start at k=0 and use the shortcut S=a/(1-r) for r<1. Funny how easy this was to understand and how a year one calc series stumped me for a minute.

[deshtom]

Would it be possible for you to show the Lebesgue integral for a more traditional function? I'd especially like to see you prove that the Riemann integral and the Lebesgue integral are equal for some function.

[AndDiracisHisProphet]

Oh no! The cantor ternary function! Dem vietnam flashbacks!
Before I even watched

[gregoryyampolsky]

This guy actually got me really excited about this question. Hes awesome

[pedro]

I love your videos and the way you explain these topics. Im a grad econ student (so my math base is not really solid) and find this really interesting. Keep it going

[legendarypokemontrainer]

You make it look so easy, excellent work Mr.peyam, waiting for more videos like this

[michaelaristidou]

Thank you sir for giving a complete and worked out example. It's hard to find examples in books. Please give more examples of more classic functions, such as x^2 or sinx, etc.

[ozzyfromspace]

I don’t yet know how to do lebesgue integrals, but I guessed the approach you used in the video. I’m surprised and happy 😊🙌🏽

[EL-eodh]

Thanks again, every enjoyable lecturer.

[tomatrix]

Wow. Lovely problem and interesting function

[jeromesnail]

I was about to make a comment about how we could solve this simply just by using symmetry, and then came the OMG way.
Kinda proud I came up with this idea by myself :)

[petohunor]

The last minute is very cool. 😊 To Riemann integral: Lets n>=0 natural numbers! The length of interval, where f(x) is 1/2**(-n) equal to length of interval, where f(x)=1-1/2**(-n). The average of f(x) these interval pairs is: (1/2**(-n) + 1 – 1(2**(-n))/2 =(1)/2 =1/2. So the integral 0 to 1 is 1/2 too.

[vkilgore]

That was definitely an 'OMG' moment at the end. Lol

[ectoplasm]

I am so glad I stuck around to the end. My jaw dropped a mile when I realised what you were about to do. So cool.

[ambidexter]

I figured out the answer after I noticed that Ramanujan sums of this function are all 1/2.

Additionally, if one draws the graph together with the graph of y=x, one can see that y=x always crosses every step of the devil's ladder exactly in the middle, leaving two equal triangles over and under the y=x line. Thus we can see that the integral is the same as the integral of y=x from 0 to 1 which is 1/2.