THE LEBESGUE INTEGRAL
The Lebesgue Integral, Simple Functions, Non-Negative Functions
Ratings 0.00 / 5.00
What You Will Learn!
- The Lebesgue Integral plays an important role in probability theory, real analysis, and many other fields in Mathematics.
- Furthermore, The Lebesgue Integral can define the integral in a completely abstract setting, giving rise to probability theory.
- Advantages of Lebesgue theory over Riemann theory: 1. Can integrate more functions (on finite intervals). 2. Good convergence theorems.
- In the study of Fourier series, Fourier transforms, and other topics, the Lebesgue integral is better able to describe how and when it is possible to take limit
Description
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. The Lebesgue integrals are the integration of functions over measurable sets, which could integrate many functions that cannot be integrated as Riemann integrals or even Riemann-Stieltjes integrals. The concept behind the Lebesgue integrals is that generally, while integrating a given function, the total area under the curve is divided into several vertical rectangles, but while determining the Lebesgue integral of the function, the area under the curve is divided into horizontal slabs, that need not be rectangles.
The Lebesgue Integral plays an important role in Probability theory, Real Analysis, and many other fields in Mathematics. In the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign (via the Monotone Convergence Theorem and Dominated Convergence Theorem).
This Impressive Course of 3 hr 7 min includes the Contents_
Introduction of Step Function
Definition of Characteristic Function & Simple Functions
The Lebesgue Integral of a Bounded Function over a set of Finite Measure.
Necessary and Sufficient Condition for a function to be Measurable.
Definition of a LEBESGUE INTEGRAL
Function that is Lebesgue Integral but not Riemann Integral.
Properties of Lebesgue Integrals.
BOUNDED CONVERGENCE THEOREM
The Integral of a Non-Negative Function
FATOU'S LEMMA
MONOTONE CONVERGENCE THEOREM
Corollary of Monotone Convergence Theorem
Definition of a Non Negative Function Integrable over the Measurable Set
Including all Propositions, Theorems and Lemma's.
Thank You.
Who Should Attend!
- A standard for a math undergraduate program, Students of MSc. mathematics , PG/UG students of maths, UGC Entrance , Under graduates, Master degree in statistics, functional analysis