Page 382 - DMTH401_REAL ANALYSIS

Real Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 32: The General Lebesgue Integral and
Convergence in Measure

CONTENTS

Objectives
Introduction
32.1 The General Lebesgue Integral
32.2 Lebesgue Convergence Theorem

32.3 Convergence in Measure
32.4 Summary
32.5 Keywords
32.6 Review Questions

32.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Explain the General Lebesgue integral of a measurable function
 Discuss the Properties of Lebesgue integral
 Discuss Lebesgue convergence theorem
 Explain Generalization of Lebesgue convergence theorem
 Describe convergence in measure of a sequence of measurable functions

Introduction

In this unit, you are going to study about the general Lebesgue integral, some of its properties,
convergence in measure and theorems related to them.

32.1 The General Lebesgue Integral

+
+
Definition: The positive part of a function f is f = f  0 i.e f (x) = max {f(x), 0}
–
The negative part of a function is f = f  0. i.e f (x) = min {f(x), 0}
–
–
+
Hence f = f – f .
And |f| = f + f –
+
+
–
Definition: A measurable function f is said to be integrable over E if f and f are both integrable
over E.
Then the integral of f is defined as
+
E ò f = E ò f - E ò f -

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