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
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