A horizontal integral?! Introduction to Lebesgue Integration

[Music] i think most of us are familiar with the usual way of calculating integrals let's consider a function f of x if i want to calculate the integral over some bound which we typically think as the area under the curve we divide the region into vertical rectangles and sum up the areas of each of these rectangles to get the final integral we consider smaller and smaller rectangles and the integral is the sum of the areas of all of these rectangles as the number of rectangles approaches infinity the sum of rectangles is called riemann sums so this technique is called riemann integration however this technique definitely has its problems the first shortcoming of the riemann integration is the difficulty to extend this intuition to higher dimensions to approximate the volume under a surface we would use cuboids rather than rectangles but if we move to four dimensions we would use well we don't know visualizing a four-dimensional object is incredibly difficult the second and most important shortcoming is some assumption of continuity take this piecewise function for example if i were to approximate the area under this curve using riemann integration how would i place a rectangle at the point of discontinuity although this can be solved by sliding the integral into 2 here's an example of something where that won't work let's define a function f x on the interval 0 comma 1 as follows f of x is equal to 1 when x is irrational and f of x is equal to 0 when x is rational now what is the integral from 0 to 1 of f of x dx this function is impossible to integrate with riemann integration if we consider one rectangle it will always have infinitely many rational numbers and infinitely many irrational numbers i think this diagram kind of gives away what the answer to this question is it appears that this graph has a lot more irrational numbers than rational numbers which means that the integral should be equal to 1. this to me was quite counter-intuitive but here's an explanation that made sense to me let's say i have a number between 0 and 1 with infinite number of decimals and let's say the first three decimals are 1 2 3 so we have 0.123. now one way for this to be rational is to repeat 1 to 3 infinitely many times the chance of the next digit being 1 is 1 and 10 or 0.1 and the chance of the next n digits being correct is 0.1 times 0.1 n times which is 0.1 to the power of n and to consider the probability of repeating this number infinitely many times we take a limit as n approaches infinity of this function 0.1 to the power of n looking at the graph of this function it's clear that this limit approaches 0. and so the probability of this number being rational is well zero so the integral has to be one still this doesn't help us with the fact that we can't integrate a function with riemann integration we need a new definition of the integral during the 20th century french mathematician henry lobeg had a clever solution to the shortcomings of the riemann integral instead of splitting up the integral along the interval or the x-axis what if we split it along the range or the y axis in other words instead of considering the values of the function in order what if we considered all the output values at once this method was beautifully summarized by the quote i showed you at the beginning of the video after sum which i've collected in my pocket i take the bills and coins out of my pocket and give them to the creditor in the order i find them until i've reached the total sum this is the riemann integral but i can proceed differently after i've taken all the money out of my pocket i ordered the bills and coins according to identical values then i pay several heaps one after the other to the creditor this is my integral think about this it intuitively solves the issue we had with the riemann integral and the irrational irrational function since we're splitting over all possible outputs we only need to consider two cases one when f of x is equal to zero and one when f of x is equal to one let's split our output into two sets and call the first set where f of x is equal to zero and x is rational a of zero and the set where f of x is equal to one and x is irrational a of one with this in mind we can write a definition for the integral from zero to one of this function think of each of the sets as covering a rectangle with length f of x and height being the size of that set and so the integral is simply zero times the size of a of zero plus one times the size of a of one the symbol mu gives the size of a set the formal term for this is the big measure i've left a few links in the description to understand what this means more so check this out if you're interested d mu tells us that this is a liebig integral as for calculating the actual values it's pretty similar to what we did before the size of a of 0 is 0 since the number of rational numbers between 0 and 1 is significantly less than the number of irrational numbers between 0 and 1. the size of a of 1 is equal to 1 since the number of irrational numbers is a lot more and so the integral is equal to 0 times 0 plus 1 times 1 which is equal to 1. a function like the one above where the function takes on a certain amount of values is called a simple function given this we can come up with a formal definition for the labeg integral of a simple function like before it's the sum of the value of the function times the number of occurrences of that value in the interval if you remember this is called the le big measure to further extend this definition we can come up with a formal definition for the labeling integral of a continuous function if you think about it a continuous function is just a simple function that takes on an infinite number of values given that we can write the big integral of a continuous function as follows the set of functions f n is a sequence of simple functions that approaches f as n approaches infinity i think it's pretty clear that the lebeg integral is powerful in terms of the variety of functions it can integrate but its abstractness weakens it quite a bit however this raises the question of why we even need such an integral it turns out that we don't deal with pure polynomials in our daily applications consider measuring an electric signal from a source it's very likely that the signal will not be set for all times and that the signal occurs at only an irrational time it also helps with generalizing a few concepts consider trying to calculate the expected value of some probabilistic event if we're dealing with a probability distribution p of x like this normal distribution the expected value is just an integral from negative infinity to positive infinity of the distribution however a lot of the times you're not dealing with a probability distribution but rather an event that only takes on a certain set of values i talked about this in my video on the coupon collectors problem here's the segment on the expected value let's say you and i are playing a coin flipping game if the coin lands heads i give you ten dollars if the coin lands tails you give me five dollars from this game we can construct this tree diagram consider each of the outcomes from your point of view if it lands tails you get 10 dollars and the chance that this is 0.5 so we weight this outcome with the value and multiply them together this results in a value of positive five similarly we lose five dollars for the other case and this also has a chance of 0.5 so if we multiply them together we get negative 2.5 and so our expected value is the sum of these two outcomes which is 2.5 this is basically a le big integral using labeg integration to define the expected value we can generalize the concept of expected value to all kinds of functions in fact the probability theory we deal with today is actually based on the idea of lobeg integration and measure theory if you're curious about some more applications three blue one brown made a video how we can apply this very irrational rational problem to a music problem so check that out if you're interested i've also left an article in the description that has some more applications before the video ends i want to talk a bit more about my patreon like i said at the beginning of the video i made one so if you want to support me you can do so i had a bit of an ethical demise making this patreon because i didn't want to feel like i was taking other people's money so only if you're in a position to do so then please support me if you're not in a financial position to support me on patreon then watching the video is enough support for me as far as rewards go i want you to think of this as a support and not as i'm paying money to get some rewards in return because i don't really know what i'm gonna give out at patreon as of right now i only have a few pretty mediocre rewards there's a couple of real unreleased animations and one unreleased video that i don't want to release anytime soon if you have any suggestions for things you want then please let me know that's about it thank you for watching [Music] [Music] bye [Music] you

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Keywords: Lebesgue integral, lebesgue measure, Lebesgue, integral, Riemann integral, Riemann integration, Lebesgue integration, manim, animation, math, vcubingx, v cubing x, vcubing x, integration, Lebesgue vs riemann, Lebesgue vs Riemann integration, lebesgue integral, lebesgue integral examples, lebesgue, measure theory

Id: LDNDTOVnKJk

Length: 9min 53sec (593 seconds)

Published: Thu Jul 09 2020