Real Analysis Part 6 (THE LEBESGUE INTEGRAL)

Name: Real Analysis Part 6 (THE LEBESGUE INTEGRAL)
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The Lebesgue Integral, Simple Functions, Non-Negative Functions, lebesgue integral, lebesgue integrals, integrals

Created byJaswinder Kaur

Last updated 7/2024

English

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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

Course content

3 sections • 18 lectures • 3h 20m total length

Introduction to Simple Functions & Characteristic Functions9:31
Explore the Lebesgue integral with respect to Lebesgue measure of a bounded function on finite measure sets, using characteristic and simple functions to build canonical representations.
Lemma on Integral of Simple Function7:17
The lemma shows that for a simple function phi = sum a_I chi_EI with disjoint measurable E_I of finite measure, the Lebesgue integral equals sum a_I measure(E_I).
Proposition on simple Functions that vanish outside a set of Finite Measure10:27
Real analysis part 6 proves linearity of the Lebesgue integral for simple functions vanishing outside a finite-measure set. If phi ≥ psi almost everywhere, then integral phi ≥ integral psi.
Sufficient Condition for 'f' to be Measurable.16:55
Demonstrate that a bounded function on a finite-measure set is measurable iff the infimum of ∫ psi equals the supremum of ∫ phi with simple phi ≤ f ≤ psi.
Necessary Condition for 'f' to be Measurable15:13
Show that if simple functions phi_n and psi_n bound f with equal infimum and supremum integrals, then define phi* and psi*; they coincide a.e., proving f is measurable.

Introduction of Step Function9:16
Explore the Lebesgue integral for bounded measurable functions on finite-measure sets, defining it via simple and step functions, infima of integrals, and the relation to the Riemann integral.
Lebesgue Integral is the Generalization of Riemann Integral20:03
shows that if f is bounded on [a,b] and Riemann integrable, then f is measurable and the Riemann integral equals the Lebesgue integral, illustrating Lebesgue's generalization.
Prove that Given Function is Lebesgue integral but not Riemann Integral5:35
Show that on the interval [0,1], f(x)=1 for rational x and f(x)=0 for irrational x is Lebesgue integrable but not Riemann integrable.
Properties of Lebesgue Integral16:40
Show linearity and additivity of the Lebesgue integral, scale by a, and equality when functions are equal almost everywhere, using simple functions with infimum and supremum arguments.
Some More Properties of Lebesgue Integral10:14
Demonstrates key Lebesgue integral properties: order and absolute value inequalities under a.e. comparison, bounds from constants, and additivity on disjoint measurable finite sets.
BOUNDED CONVERGENCE THEOREM10:08
The bounded convergence theorem states that measurable f_n on a finite-measure set E with |f_n| ≤ m converge to f, so Lebesgue integral over E equals limit of ∫_E f_n.

The Integral of a Non Negative Function18:59
learn how to define the Lebesgue integral of a nonnegative measurable function on a measurable set, prove linearity for nonnegative functions, and establish monotonicity with respect to almost everywhere inequalities.
FATOU'S LEMMA10:41
Explore Fatou's lemma: nonnegative measurable f_n converge almost everywhere to f, yielding a Lebesgue integral inequality, with the bounded convergence theorem used in the proof.
MONOTONE CONVERGENCE THEOREM11:16
Apply the monotone convergence theorem to an increasing sequence of non-negative measurable f_n converging to f, showing the Lebesgue integral of f equals the limit of the integrals of f_n.
Corollary of Monotone Convergence Theorem11:06
Demonstrates the corollary of the monotone convergence theorem: for non-negative measurable u_n with f = sum u_n, the Lebesgue integral of f equals the sum of the integrals of u_n.
Proposition of non negative function for Disjoint sequence of Measurable Sets3:22
Show that for a nonnegative function f and a disjoint sequence of measurable sets e_I with union E, Lebesgue integral over E equals sum of integrals over e_I.
Definition of Non Negative Measurable Function Integrable over E4:40
Show that nonnegative measurable f and g with f integrable on E and f > g imply g integrable and ∫_E (f − g) = ∫_E f − ∫_E g.
Proposition of Non Negative Function 'f' which is Integrable over a set E9:26
Establishes an epsilon-delta property for nonnegative integrable f on E: every small-measure subset A has integral less than epsilon, proven using monotone convergence and bounded case analysis.

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 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 this course is for:

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

Real Analysis Part 6 (THE LEBESGUE INTEGRAL)

What you'll learn

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Course content

The Lebesgue Integral INTRODUCTION5 lectures • 59min

INTRODUCTION of Lebesgue Integral6 lectures • 1hr 12min

The Integral of a NON NEGATIVE FUNCTION7 lectures • 1hr 10min

Requirements

Description

Who this course is for: