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
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
(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
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,
c1f(x) + c2g(x) dx = c1
f(x) dx + c2
In particular, this implies that
0 dx = 0.