Page 107 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Richa Nandra, Lovely Professional University

Notes Unit 9: Measure Spaces

CONTENTS
Objectives
Introduction

9.1 Measure Space
9.1.1 Null Set in a Measure Space
9.1.2 Complete Measure Space

9.1.3 Measurable Mapping
9.2 Summary
9.3 Keywords
9.4 Review Questions
9.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define measure space.

 Define null set in a measure space.
 Understand theorems based on measure spaces.
 Solve problems on measure spaces.

Introduction

A measurable space is a set S, together with a non-empty collection, S, of subsets of S, satisfying
the following two conditions:
1. For any A, B in the collection S, the set A – B is also in S.
1
2. For any A , A , … S, A S.
1 2 i
The elements of S are called measurable sets. These two conditions are summarised by saying
that the measurable sets are closed under taking finite differences and countable unions.

9.1 Measure Space

Measurable Space: Let  be a -algebra of subsets of set X. The pair (X, ) is called a measurable
space. A subset E of X is said to be -measurable if E .
(a) If is a measure on a -algebra  of subsets of a set X, we call the triple (X, , u) a measure
space.

(b) A measure on a -algebra  of subsets of a set X is called a finite measure if m (X) < . In
this case (X, , ) is called a finite measure space.

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