2 1. The Regulated and Riemann Integrals

There are several properties which we want an integral to satisfy

no matter how we define it. It is worth enumerating them at the be-

ginning. We will need to check them and refine them for our different

definitions.

1.2. Basic Properties of an Integral

We will consider the value of the integral of functions in various col-

lections. These collections all have a common domain which, for our

purposes, is a closed interval. They are also closed under the opera-

tions of addition and scalar multiplication. Such a collection is a vec-

tor space of real-valued functions (see, for example, Definition A.9.1).

More formally, recall that a non-empty set of real-valued functions

V defined on a fixed closed interval is a vector space of functions

provided:

(1) If f, g ∈ V, then f + g ∈ V.

(2) If f ∈ V and c ∈ R, then cf ∈ V.

Notice that this implies that the constant function 0 is in V. All of the

vector spaces we consider will contain all of the constant functions.

Three simple examples of vector spaces of functions defined on

some closed interval I are the constant functions, the polynomial

functions, and the continuous functions.

An “integral” defined on a vector space of functions V is a way

to assign a real number to each function in V and each subinterval

of I. For the function f ∈ V and the subinterval [a, b] we denote this

value by

b

a

f(x) dx and call it “the integral of f from a to b.”

All the integrals we consider will satisfy five basic properties

which we now enumerate.

I. Linearity: For any functions f, g ∈ V, any a, b ∈ I, and

any real numbers c1, c2,

b

a

c1f(x) + c2g(x) dx = c1

b

a

f(x) dx + c2

b

a

g(x) dx.

In particular, this implies that

b

a

0 dx = 0.