Page 113 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Richa Nandra, Lovely Professional University

Notes Unit 10: Measurable Functions

CONTENTS
Objectives
Introduction

10.1 Measurable Functions
10.1.1 Lebesgue Measurable Function/Measurable Function
10.1.2 Almost Everywhere (a.e.)

10.1.3 Equivalent Functions
10.1.4 Non-Negative Functions
10.1.5 Characteristic Function
10.1.6 Simple Function
10.1.7 Step Function

10.1.8 Convergent Sequence of Measurable Function
10.1.9 Egoroff's Theorem
10.1.10 Riesz Theorem

10.2 Summary
10.3 Keywords
10.4 Review Questions
10.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand measurable functions.
 Define equivalent functions and characteristic function.

 Describe Egoroff's theorem and Riesz theorem.
 Define simple function and step function.
Introduction

In this unit, we shall see that a real valued function may be Lebesgue integrable even if the
function is not continuous. In fact, for the existence of a Lebesgue integral, a much less restrictive
condition than continuity is needed to ensure integrability of f on [a, b]. This requirement gives
rise to a new class of functions, known as measurable functions. The class of measurable functions
plays an important role in Lebesgue theory of integration.

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