Page 309 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 26: Lebesgue Measure

Unit 26: Lebesgue Measure Notes

CONTENTS
Objectives
Introduction
26.1 Lindelof’s Theorem
26.2 Lebesgue Outer Measure
26.3 Non-measurability

26.4 Measurable Sets and Lebesgue Measure
26.5 Step Functions and Simple Functions
26.6 Measurable Functions
26.7 Summary
26.8 Keywords
26.9 Review Questions
26.10 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss the definition of outer measure of sets
 Define outer measure of an interval
 Explain some important properties of outer measure
 Define measurable sets
 Describe measure of countable union of measurable sets
 Measure countable intersection of measurable sets

Introduction

In last unit you have studied about mean value theorems of Riemann Stieltjes integral. In this
unit we are going to study about Lebesgue outer measure of a set, measurable sets and Lebesgue
measure, their important properties.

We know that the length of an interval is defined to be the difference between two end points.
In this unit, we would like to extend the idea of “length” to arbitrary (or at least, as many as
possible) subsets of . To begin with, let’s recall two important results in topology.

26.1 Lindelof’s Theorem

Proposition: Every open subset V of  is a countable union of disjoint open intervals.
Proof: For each x  V, there is an open interval I with rational endpoints such that x  I  V. Then
x x
the collection {I } is evidently countable and
x xV
V =  I .
x

x V

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