Page 165 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 13: Signed Measures

CONTENTS
Objectives
Introduction

13.1 Signed Measures
13.1.1 Signed Measure: Definition
13.1.2 Positive Set, Negative Set and Null Set

13.1.3 Hahn Decomposition Theorem
13.1.4 Hahn Decomposition: Definition
13.2 Summary
13.3 Keywords
13.4 Review Questions

13.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define signed measure.
 Describe positive and negative and null sets.
 Solve problems on signed measure.

Introduction

We have seen that a measure is a non-negative set function. Now we shall assume that it takes
both positive and negative values. Such assumption leads us to a new type of measure known as
signed measure. In this unit, we shall start with definition of signed measure and we shall prove
some important theorems on it.

13.1 Signed Measures

13.1.1 Signed Measure: Definition

Definition: Let the couple (X, ) be a measurable space, where  represents a -algebra of
subsets of X. An extended real valued set function
:  [– , ]

defined on  is called a signed measure, if it satisfies the following postulates:
(i) assumes at most one of the values – or + .
(ii) ( ) = 0.

158 LOVELY PROFESSIONAL UNIVERSITY