Page 347 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 29: The Lebesgue Integral of Bounded Functions

Unit 29: The Lebesgue Integral of Bounded Functions Notes

CONTENTS
Objectives
Introduction

29.1 Simple Functions Vanishing Outside a Set of Finite Measure
29.2 Properties of the Lebesgue Integral
29.3 Bounded Measurable Functions Vanishing Outside a Set of Finite Measure

29.4 Summary
29.5 Keyword
29.6 Review Questions
29.7 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the Lebesgue integral of bounded functions over a set of finite measure
 Explain properties of the Lebesgue integral of bounded functions over a set of finite
measure
 Describe bounded convergence theorem

Introduction

After getting basic knowledge of the Lebesgue measure theory, we now proceed to establish the
Lebesgue integration theory.
In this unit, unless otherwise stated, all sets considered will be assumed to be measurable.

We begin with simple functions.

29.1 Simple Functions Vanishing Outside a Set of Finite Measure

Recall that the characteristic function  for any set A is defined by
A
Î
1 if x A
 (x) =
A { 0 otherwise
A function  : E  is said to be simple if there exists a , a ,...., a Î  and E , E ,...., E  E such
1 2 n 1 2 n
that  = å n i 1 a  i E . Note that here the E ’s are implicitly assumed to be measurable, so a simple
i
=
i
function shall always be measurable. We have another characterization of simple functions:
Proposition: A function  : E  is simple if and only if it takes only finitely many distinct
–1
values a , a , .... a and  {a } is a measurable set for all i = 1, 2,....., n.
1 2 n i

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