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Richa Nandra, Lovely Professional University Unit 11: Integration

Unit 11: Integration Notes

CONTENTS
Objectives
Introduction

11.1 Integration
11.1.1 The Riemann Integral
11.1.2 Lebesgue Integral of a Bounded Function over a Set of Finite Measure

11.1.3 The Lebesgue Integral of a Non-negative Function
11.1.4 The General Lebesgue Integral
11.2 Summary
11.3 Keywords
11.4 Review Questions

11.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define the Riemann integral and Lebesgue integral of bounded function over a set of
finite measure.
 Understand the Lebesgue integral of a non-negative function.

 Solve problems on integration.
Introduction

We now come to the main use of measure theory: to define a general theory of integration. The
particular case of the integral with respect to the Lebesgue measure is not, in any way, simpler
the general case, which will give us a tool of much wider applicability.

11.1 Integration

11.1.1 The Riemann Integral

Let f be a bounded real valued function defined on the interval [a, b] and let a = x < x < … < x = b
0 1 n
be a sub-division of [a, b].
Then for each sub-division we can define the sums

n
S = (x i x ) M i
i 1
i 1
n
and s = (x i x ) m i
i 1
i 1

LOVELY PROFESSIONAL UNIVERSITY 121