Chapter 1

The Regulated and

Riemann Integrals

1.1. Introduction

This text is devoted to exploring the definition and properties of the

definite integral. We will consider several different approaches to

defining the definite integral

b

a

f(x) dx

of a function f : [a, b] → R. These definitions will all assign the

same value to the definite integral, but they differ in the size of the

collection of functions to which they apply. For example, we might try

to evaluate the Riemann integral (the ordinary integral of beginning

calculus) of the function

f(x) =

0, if x is rational;

1, otherwise.

The Riemann integral

1

0

f(x) dx is, as we will see, undefined; but

the Lebesgue integral, which we will develop, has no diﬃculty with f

and indeed

1

0

f(x) dx = 1.

1

http://dx.doi.org/10.1090/stml/048/01