The Lebesgue Integral

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