Page 336 - DMTH401_REAL ANALYSIS

Real Analysis Richa Nandra, Lovely Professional University

Notes Unit 28: Sequences of Functions and
Littlewood’s Third Principle

CONTENTS
Objectives
Introduction
28.1 Limsups and Liminfs of Sequences
28.2 Operations on Measurable Functions

28.3 Littlewood’s Third Principle
28.4 Summary
28.5 Keywords
28.6 Review Questions
28.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss the limsups and liminfs of sequences
 Describe operations on measurable functions

 Explain Littlewood's third principle
Introduction

In this unit we continue our study of measurability. We show that measurable functions are very
robust in the sense that they are closed under just about any kind of arithmetic or limiting
operation that you can imagine: addition, multiplication, division,…, and most importantly,
they are closed under just about any conceivable limiting process. We also discuss Littlewood’s
third principle on limits of measurable functions.

28.1 Limsups and Liminfs of Sequences

Before discussing limits of sequences of functions we need to start by talking about limits of
sequences of extended real numbers.
For a sequence {a } of extended real numbers, we know, in general, that lim a does not exist; for
n n
example, it can oscillate such as the sequence. However, for the sequence, assuming that the
sequence continues the way it looks like it does, it is clear that although limit lim a does not
n
exist, the sequence does have an “upper” limiting value, given by the limit of the odd-indexed
a ’s and a “lower” limiting value, given by the limit of the even-indexed a ’s. Now, how do we
n n
find the “upper” (also called “supremum”) and “lower” (also called “infimum”) limits of {a }? It
n
turns out there is a very simple way to do so, as we now explain.

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