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Sachin Kaushal, Lovely Professional University Unit 31: The Integral of a Non-negative Function

Unit 31: The Integral of a Non-negative Function Notes

CONTENTS
Objectives
Introduction

31.1 Integration of Non-negative Measurable Functions
31.2 Extended Real-valued Integrable Functions
31.3 Summary

31.4 Keywords
31.5 Review Questions
31.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss the integral of a non-negative function
 Explain Properties of the integral of non-negative functions
 Describe Monotone convergence theorem and
 Definition of Integrable function over a measurable set

Introduction

In this unit we are going to study about the definition and the properties of the integral of non-
negative functions and some important theorems.

31.1 Integration of Non-negative Measurable Functions

We integrate non-negative measurable functions through approximation by bounded measurable
functions vanishing outside a set of finite measure, which we studied earlier.

Definition: For a non-negative measurable function f : E  [0, ] (where E is a set which may be
of finite or infinite measure), we define

:
A ò f = sup{ A ò j j £ f on A, j Î B (E) }
0
for any A E.
Note that for non-negative bounded measurable functions vanishing outside a set of finite
measure, this definition agrees with the old one. Also note that we allow the functions to take
infinite value here.
We verify the monotonicity and linearity of such integrals.
Proposition: Suppose f, g : E  [0, ] are non-negative measurable and A  E.

(a) If f £ g a.e. on A then ò A f £ A ò g .
(b) For  > 0, f + g and f are non-negative measurable functions too and

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