The Lebesgue Integral

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The Lebesgue Integral
Published 5/2023
MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
Language: English | Size: 750.53 MB | Duration: 3h 21m

The Lebesgue Integral, Simple Functions, Non-Negative Functions

What you'll 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
Requirements
Basic knowledge of Riemann Integration and Basic concepts of sequence and series with Limits.
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 FunctionDefinition of Characteristic Function & Simple FunctionsThe 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 INTEGRALFunction that is Lebesgue Integral but not Riemann Integral.Properties of Lebesgue Integrals.BOUNDED CONVERGENCE THEOREMThe Integral of a Non-Negative FunctionFATOU'S LEMMAMONOTONE CONVERGENCE THEOREM Corollary of Monotone Convergence TheoremDefinition of a Non Negative Function Integrable over the Measurable SetIncluding all Propositions, Theorems and Lemma's.Thank You.
Overview
Section 1: The Lebesgue Integral INTRODUCTION
Lecture 1 Introduction to Simple Functions & Characteristic Functions
Lecture 2 Lemma on Integral of Simple Function
Lecture 3 Proposition on simple Functions that vanish outside a set of Finite Measure
Lecture 4 Sufficient Condition for 'f' to be Measurable.
Lecture 5 Necessary Condition for 'f' to be Measurable
Section 2: INTRODUCTION of Lebesgue Integral
Lecture 6 Introduction of Step Function
Lecture 7 Lebesgue Integral is the Generalization of Riemann Integral
Lecture 8 Prove that Given Function is Lebesgue integral but not Riemann Integral
Lecture 9 Properties of Lebesgue Integral
Lecture 10 Some More Properties of Lebesgue Integral
Lecture 11 BOUNDED CONVERGENCE THEOREM
Section 3: The Integral of a NON NEGATIVE FUNCTION
Lecture 12 The Integral of a Non Negative Function
Lecture 13 FATOU'S LEMMA
Lecture 14 MONOTONE CONVERGENCE THEOREM
Lecture 15 Corollary of Monotone Convergence Theorem
Lecture 16 Proposition of non negative function for Disjoint sequence of Measurable Sets
Lecture 17 Definition of Non Negative Measurable Function Integrable over E
Lecture 18 Proposition of Non Negative Function 'f' which is Integrable over a set E
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

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