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.
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