Page 324 - DMTH401_REAL ANALYSIS

Real Analysis Richa Nandra, Lovely Professional University

Notes Unit 27: Measurable Functions and Littlewood’s
Second Principle

CONTENTS
Objectives
Introduction
27.1 Measurable Functions
27.2 Measurability and Continuity
27.3 Littlewood’s Second Principle
27.4 Borel Measurability on Topological Spaces
27.5 The Concept of Almost Everywhere
27.6 Summary
27.7 Keywords
27.8 Review Questions
27.9 Further Readings

Objectives

After studying this unit, you will be able to:

 Define measurable functions
 Discuss the Sum, difference; scalar product and product of measurable functions are
measurable
 Explain Littlewood's Theorems

Introduction

In this unit we study the concept of measurability. We shall see that measurable functions are
basically very robust (or strong or durable) continuous-like functions. We make “continuous-like”
precise in Luzin’s Theorem, which is where Littlewood got his second principle. We also study
the concept of almost everywhere.

27.1 Measurable Functions

A measurable space is a pair (X, S ) where X is a set and, S is a -algebra of subsets of X. The
elements of, S are called measurable sets. Recall that a measure space is a triple (X, S , ) where
 is a measure on S ; if we leave out the measure we have a measurable space.
In the discussion at the beginning of this unit we saw that in order to define the integral of a
function f : X   , we needed to require that

f (I)  S for each I  S and f (a, ]  S for each a  .
–1
–1
If these properties hold, we say that f is measurable. It turns out that we can omit the first
condition because it follows from the second. Indeed, since
–1
–1
–1
f (a, b] = f (a, ]\f (b, ],

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