Measure theory | Definitions of mathematical integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the necessary abstractions for this. The Lebesgue integral plays an important role in probability theory, real analysis, and many other fields in mathematics. It is named after Henri Lebesgue (1875–1941), who introduced the integral. It is also a pivotal part of the axiomatic theory of probability. The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of integration of a function defined on a sub-domain of the real line with respect to the Lebesgue measure. (Wikipedia).
In this video, I present an overview (without proofs) of the Lebesgue integral, which is a more general way of integrating a function. If you'd like to see proods of the statements, I recommend you look at fematika's channel, where he gives a more detailed look of the Lebesgue integral. In
From playlist Real Analysis
Measure Theory 2.1 : Lebesgue Outer Measure
In this video, I introduce the Lebesgue outer measure, and prove that it is, in fact, an outer measure. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
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 W
From playlist Real Analysis
In this video, I show how to calculate the integral of x^3 from 0 to 1 but using the Lebesgue integral instead of the Riemann integral. My hope is to show you that they indeed produce the same answer, and that in fact Riemann integrable functions are also Lebesgue integrable. Enjoy!
From playlist Real Analysis
A horizontal integral?! Introduction to Lebesgue Integration
Support me on Patreon! https://patreon.com/vcubingx Join my discord server! https://discord.gg/Kj8QUZU Terry Tao's book on Measure Theory: https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf Learn about Lebesgue Measure: https://www.math.ucdavis.edu/~hunter/measure_
From playlist Analysis
Measure Theory 3.1 : Lebesgue Integral
In this video, I define the Lebesgue Integral, and give an intuition for such a definition. I also introduce indicator functions, simple functions, and measurable functions.
From playlist Measure Theory
Riemann-Integral vs. Lebesgue-Integral
English version here: https://www.youtube.com/watch?v=PGPZ0P1PJfw Unterstützt den Kanal auf Steady: https://steadyhq.com/en/brightsideofmaths Ihr werdet direkt informiert, wenn ich einen Livestream anbiete. Hier erkläre ich den Unterschied zwischen Riemann-Integral und Lebesgue-Integral
From playlist Analysis
Think you can integrate every function? Unfortunately you can't, but it's not your fault! In this case, it's the math that's wrong and needs to be updated (just like sometimes you update your operating system). This video presents the limit of Riemann (calculus-style) integration and the b
From playlist Integration
Measure Theory 3.3 : Riemann Integral Equals Lebesgue Integral
In this video, I describe a new way of defining the Riemann Integral, and use that to prove that the Riemann and Lebesgue Integrals are the same for Riemann Integrable functions. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
Measure Theory - Part 6 - Lebesgue integral
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From playlist Measure Theory
Lec 8 | MIT 6.450 Principles of Digital Communications I, Fall 2006
Lecture 8: Measure, fourier series, and fourier transforms View the complete course at: http://ocw.mit.edu/6-450F06 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.450 Principles of Digital Communications, I Fall 2006
Real Analysis - Eva Sincich - Lecture 01
From playlist Machine learning
5 Levels of Integration | Nathan Dalaklis
Area, Riemann Sums, Integration formulas, Riemann integrability, and Lebesgue/Measure Theoretic Integrability are the same mathematical beast through different lenses at different levels of mathematical understanding and rigor. Usually you'd spend several years learning about these topics,
From playlist The New CHALKboard
Measure Theory 3.4: Monotone Convergence Theorem
In this video, I will be proving the Monotone Convergence Theorem for Lebesgue Integrals. Email : fematikaqna@gmail.com Subreddit : https://www.reddit.com/r/fematika Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
Integration 6 The Fundamental Theorem of Calculus
Explanation of the fundamental theorem of calculus in an easy to understand way.
From playlist Integration
In this video, I prove the famous Riemann-Lebesgue lemma, which states that the Fourier transform of an integrable function must go to 0 as |z| goes to infinity. This is one of the results where the proof is more important than the theorem, because it's a very classical Lebesgue integral
From playlist Real Analysis