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Sachin Kaushal, Lovely Professional University Unit 30: Riemann's and Lebesgue

Unit 30: Riemann's and Lebesgue Notes

CONTENTS
Objectives
Introduction

30.1 Riemann vs. Lebesgue
30.2 Small Subsets of  d
30.3 About Functions Behaving Nicely Outside a Small Set

30.4 -algebras and Measurable Spaces
30.5 Summary
30.6 Keywords
30.7 Review Questions
30.8 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss Riemann's and Lebesgue
 Explain the small subsets of R
 Discuss the functions outside small set

Introduction

In last unit you have studied about the Lebesgue integral of bounded functions. In this unit we
are going to study about the definition and the difference of Riemann's and Lebesgue.

30.1 Riemann vs. Lebesgue

Measure theory helps us to assign numbers to certain sets and functions to a measurable set we
may assign its measure, and to an integrable function we may assign the value of its integral.
Lebesgue integration theory is a generalization and completion of Riemann integration theory.
In Lebesgue’s theory, we can assign numbers to more sets and more functions than what is
possible in Riemann’s theory. If we are asked to distinguish between Riemann integration
theory and Lebesgue integration theory by pointing out an essential feature, the answer is
perhaps the following.
Riemann integration theory  finiteness.
Lebesgue integration theory  countable infiniteness.

Riemann integration theory is developed through approximations of a finite nature (e.g.: one
2
tries to approximate the area of a bounded subset of  by the sum of the areas of finitely many
rectangles), and this theory works well with respect to finite operations – if we can assign numbers
to finitely many sets A ,…, A and finitely many functions f ,…,f , then we can assign numbers
1 n 1 n
to A A , f ++ f , max{f ,…, f }, etc. The disadvantage of Riemann integration theory is
1 n 1 n 1 n
that it does not behave well with respect to operations of a countably infinite nature - there may

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