Measure Theory 6 | Lebesgue Integral

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This is part 6 of 22 videos.

#MeasureTheory
#Analysis
#Integral
#Calculus
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I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[tanajkamheangpatiyooth]

This measure theory series is beneficial to humanity. Thank you for making it. After an exhaustive search, this is the best explanation for Lebesgue integral.

[jui-huichung]

I love how you fast forward the writing to save time. Great!

[harryliu]

This series is so helpful. I’m trying to understand probability theory rigorously. Your video and some of my own reading just help me realize that random variables are nothing but measurable functions. This has transformed how I look at probabilities in general. Before I always thought there was some randomness involved. Now Using the concept of measure, I feel like the likelihood of events can be categorized by measure. This is a static global perspective, and it just makes everything easier to think about and work with. Very exciting stuff to learn!

[PunmasterSTP]

Lebesgue? More like "These videos are the best!" Thanks again for making and sharing such a high-quality resource.

[sjoerdglaser]

I am going to follow a few lectures about Lebesgue integrals. A friend of mine who is good at maths warned me it will be very difficult. These videos give me a very clear idea and intuitions about what I will be learning. It gives me a framework everything will fit in hopefully. Thank you so much!

[B_r_i_t_t]

These videos are great, usually when I'm watching math videos on youtube they are quite old. I was shocked when I saw how recently these have come out. Looking forward to more!

[tron_tler]

Thank you so much. You're a life saver. My professor just sent me notes, no lecture, and I had a hard time understanding her notes, until I saw your videos. Thank you.

[springvibes]

I have an exam of Real Analysis I for thesis qualification, and you just give me a brief understanding of measure. Wish me luck cuz I really did not get familiar with proofs yet, and already no much time left.

Thank you, your videos really helpful.

[shuklakuldeep]

These are the best videos on the measure theory one can make.
It would be great if you could make more videos on other areas of mathematics like TOPOLOGY, REAL ANALYSIS

[angelmendez-rivera]

For those who would like to see the connection between the Riemann integral and the measure-theoretic integrals, I am going to go back to some definitions, and analyze the fundamental mathematical object underlying both definitions.

Consider a function f : [a, b] —> R. A tag on the interval [a, b] is an n-tuple t : {m : 0 =< m =< n} —> [a, b], where if i < j, then t(i) < t(j). If such a tag t is defined, then the sum of f[t(i)]·w(i) is well-defined. Here, w is an n-tuple of real numbers, this time not constrained to the interval [a, b], and this is called a weight. This is the fundamental object of study. This sum is called a w-weighted sum of f over the tag t.

Now, consider the set of closed subintervals of [a, b], denoted c([a, b]), and consider a map p : range(t) —> c([a, b]), with the consideration that the intersection of any two closed intervals in c([a, b]) is either the common endpoint of both intervals, or the empty set, with the consideration that the union of the range of p is [a, b], and with the consideration that t(i) is an element of the closed interval p[t(i)]. This turns p into an interval-partition with a tag t, and the endpoints of these intervals are denoted x(i). If we define w(i) := x(i + 1) – x(i), then this special case of the sum of f[t(i)]·w(i) is called a Riemann sum of f. Riemann sums of f are what we use to define the Riemann integral of f.

When considering the interval-partition p, it may be more general to consider w(i) = μ(p[t(i)]), instead of w(i) := x(i + 1) – x(i). The idea here is that p[t(i)] is equal to a closed interval, whose value depends on the value of t(i), and the endpoints of this interval are x(i) and x(i + 1). μ can be interpreted to be a "size function" at the intuitive level, and so the "standard" size of an interval p[t(i)], with endpoints x(i + 1) and x(i), is x(i + 1) – x(i). In this generalization, one can allow for flexibility as to what μ is, as to not be restricted to the standard size of a closed interval. So the sum that you obtain from this is different from a Riemann sum, and is instead equal to f[t(i)]·μ(p[t(i)]), which you may call a μ-sum. The generalization can now have for f[t(i)] replaced by a lower bound of the set sp(f, p) := {f[t(i)] : x(i) =< t(i) =< x(i + 1)}. This lower bound can simply be called c(i), so the weighted sum is the sum of c(i)·μ(p[t(i)]). Now, this is really starting to resemble symbolically the kind of weigted sum that the measure-theoretic integrals are defined by, if μ is interpreted as being a measure, or measure-like. Finally, some generalizations can be made here: rather than defining p to assign a closed subinterval of [a, b] to every point of the tag t, p can simply assign a measurable set S(i) to every t(i), so long as the union of S(i) is still equal to [a, b], and so long as the pairwise intersection of S(i) is a null set. Furthermore, the domain of f allowed, rather than being a closed interval [a, b], can be generalized to simply be a set X, with (X, A) being a measurable space, and f being a measurable function. It is the supremum of the set of every such sum in this generalization that is equal to the μ-integral.

So what is the Riemann integral with respect to this μ-theory? It is an integral with respect to μ, where μ is specifically the Lebesgue measure, restricted to closed intervals, and with c(i) := f[t(i)], and with the added restriction that it only is the integral of f if the infimum of the aforementioned set, where c(i) is instead an upper bound of sp(f, p), rather than a lower bound. The gauge integral is also defined using these weighted sums of functions over tags, together with restrictions as well.

[pauljohncapote]

I met this topic in Real Analysis during my MS yet I can't remember anything. Now that I am taking my PhD I am starting to appreciate it. I am majoring Statistics but Real Analysis is a corequisite for Probability Theory. Please make lectures on Theory of Probabilities. Thank you.

[byoungsoolee]

아주 차분하고, 독자 입장에서 잘 설계된 고급 수학 강의를 들으니 참 좋습니다. 배우고 갑니다. 감사합니다.

[haggaisimon]

This lecture I like the most so far. I see now why they use positive step functions. I knew that they use positive functions but I didn’t know why.

[johnstroughair]

Amazing, I think I finally understood what a Lebesgue Integral is!

[willmurphy]

Such a good video at understanding the reasoning behind how we are approximating the integral. I feel like many books/videos don't do a good job at providing the reasoning behind taking the supremum. Thanks for this!

[mastershooter]

wow that was very cool how lebesgue integral is basically like riemann integration tipped on it's side (plus step functions)

[yearsoldboomer]

Thx a lot you help me a lot to understand how it all waorks, I ve watched all of your videos of measure theory and now I am the one who teach my friends what measure theroy is :D

[kiranboddeda]

Some are tough to believe but we need to believe and learn further...I imagine always how do I proceed further when one thing is not yet clear. But its a nice feeling of the subject!

[varnita]

Thank you very much for best explanation in this series.

[YorkiePP]

Thank you for these videos, they're the only thing keeping me afloat in my measure theory course <3