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!
Note: Photo credit goes to Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method - Scientific Figure on ResearchGate. Available from: www.researchgate.net/For-aln2...

[@meccamiles7816]

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

[@joseantonioghz4324]

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.

[@snakefang1863]

When I saw the staircase-looking graph, I went for a more geometric approach:

[@pedro9101994]

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

[@michaelaristidou2605]

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.

[@diegoostoja-kowalski5551]

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

[@EL-eo8dh]

Thanks again, every enjoyable lecturer.

[@AndDiracisHisProphet]

Oh no! The cantor ternary function! Dem vietnam flashbacks!

[@legendarypokemontrainer1995]

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

[@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 😊🙌🏽

[@kurtu5]

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.

[@Zonnymaka]

Very nice! Good job DrPi

[@gregoryyampolsky518]

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

[@tomatrix7525]

Wow. Lovely problem and interesting function

[@vkilgore11]

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

[@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.

[@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.

[@petohunor6737]

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.

[@fatimaalhoti7804]

Thanks for your help 🙏