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Richa Nandra, Lovely Professional University Unit 12: General Convergence Theorems

Unit 12: General Convergence Theorems Notes

CONTENTS
Objectives
Introduction
12.1 General Convergence Theorems
12.1.1 Convergence almost Everywhere

12.1.2 Pointwise Convergence
12.1.3 Uniform Convergence, Almost Everywhere (a.e.)
12.1.4 Bounded Convergence Theorem
12.1.5 Fatou’s Lemma
12.1.6 Monotone Convergence Theorem
12.1.7 Lebesgue Dominated Convergence Theorem
12.2 Summary

12.3 Keywords
12.4 Review Questions
12.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Understand bounded convergence theorem.

 State and prove monotone convergence theorem and Lebesgue dominated convergence
theorem.
 Solve related problems on these theorems.

Introduction

Convergence of a sequence of functions can be defined in various ways and there are situations
in which each of these definitions is natural and useful. In this unit, we shall study about
convergence almost everywhere, pointwise and uniform convergence. We shall also prove
bounded convergence theorem and monotone convergence theorem which are so useful in
solving problems on convergence. The dominated convergence theorem is one of the most
important results of Lebesgue’s integration theory. It gives a general sufficient condition for the
validity of proceeding to the limit of a sequence of functions under the integral sign. It is an
invaluable tool to study functions defined by integrals.

12.1 General Convergence Theorems

12.1.1 Convergence almost Everywhere

Let <f > be a sequence of measurable functions defined over a measurable set E. Then <f > is said
n n
to converge almost everywhere in E if there exists a subset E of E s.t.
0

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