Lecture

Lebesgue Integral: Comparison with Riemann

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

MATH-205: Analysis IV - Lebesgue measure, Fourier analysis

Learn the basis of Lebesgue integration and Fourier analysis ...

Description

This lecture discusses the comparison between Lebesgue and Riemann integrals, showing that when the Riemann integral exists, it is the same as the Lebesgue integral. The instructor explains the concept through various examples and theorems, such as Fubini's theorem and the Tonelli theorem. The lecture also covers the practical application of exchanging the order of integration and computing integrals in different scenarios.

Instructor

Maria Colombo

Teaching & PhD PhD Students Roberto Colombo, Antonio Tirotta, Carl Johan Peter Johansson, Giulia Mescolini Past EPFL PhD Students Luigi De Rosa, Silja Noëmi Aline Haffter, Massimo Sorella Courses Analysis IV - Lebesgue measure, Fourier analysis MATH-205 Learn the basis of Lebesgue integration and Fourier analysis Topics on the Euler and Navier-Stokes equations MATH-647 This topics course focuses on recent and classical fundamental results on the Euler and Navier-Stokes equa

Official source

Ontological neighbourhood

Mathematics

Analysis: Measure theory

Related concepts (32)

Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo Integration.

Improper integral

In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not continuous functions.

Lebesgue 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.

Fubini's theorem

In mathematical analysis, Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands.

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.

Related lectures (64)

Analysis IV: Convergence Theorems and Integrable Functions

Covers convergence theorems and integrable functions, including the Lebesgue integral and Borel-Cantelli sets.

Improper Integrals: Convergence and Comparison

Explores improper integrals, convergence criteria, comparison theorems, and solid revolution.

Lebesgue Integral: Criteria and Analysis

Explores the concept of Lebesgue integrability and the criteria for Lebesgue integrability, emphasizing the importance of upper and lower integrals.

Lebesgue Integration: Simple Functions

Covers the Lebesgue integration of simple functions and the approximation of nonnegative functions from below using piecewise constant functions.

Fubini Theorem on Closed Rectangles