Spectral Theory in Hilbert Space

124 B. STIELTJES INTEGRALS
Definition B.5. Let f be a real-valued function defined on [a, b].
Then the total variation of f over [a, b] is
(B.1) V (f) = sup
∆
n
|f(xk) − f(xk−1)|,
k=1
the supremum taken over all partitions ∆ = {x0, x1, . . . , xn} of [a, b].
We have 0 ≤ V (f) ≤ +∞, and if V (f) is finite, we say that f has
bounded variation on [a, b].
When the interval considered is not obvious from the context, one
may write the total variation of f over [a, b] as V b
a (f); another common
notation is b
|df|. As we mentioned above, a function of bounded
a
variation can also be characterized as a function which is the difference
of two non-decreasing functions.
Theorem B.6.
(1) The total variation V b
a (f) is an interval additive function, i.e.,
if a < x < b we have V x
a (f) + V b
x (f) = V b
a (f).
(2) A function of bounded variation on an interval [a, b] may be
written as the difference of two non-decreasing functions. Conversely,
any such difference is of bounded variation.
(3) If f is of bounded variation on [a, b], then there are nondecreasing
functions P and N, such that f(x) = f(a)+P (x)−
N(x), called the positive and negative variation functions of f
on [a, b], with the following property: For any pair of nondecreasing
functions u, v for which f = u − v holds u(x) ≥
u(a) + P (x) and v(x) ≥ v(a) + N(x) for a ≤ x ≤ b.
Proof. It is clear that if a < x < b and ∆, ∆ ′ are partitions of
[a, x] respectively [x, b], then ∆ ∪ ∆ ′ is a partition of [a, b]; the cor-
responding sum is therefore ≤ V b
a (f). Taking supremum over ∆ and
then ∆ ′ it follows that V x
a (f) + V b
x (f) ≤ V b
a (f). On the other hand,
in calculating V b
a (f), we may restrict ourselves to partitions ∆ con-
taining x, since adding new points can only increase the sum (B.1). If
∆ = {x0, . . . , xn} and x = xp we have p
k=1 |f(xk) − f(xk−1)| ≤ V x
a (f)
respectively m
k=p+1 |f(xk) − f(xk−1)| ≤ V b
x (f). Taking supremum over
all ∆ we obtain V b
a (f) ≤ V x
a (f) + V b
x (f). The interval additivity of the
total variation follows.
Setting T (x) = V x
a (f) the function T is finite in [a, b]; it is called
the total variation function of f over [a, b]. Since by interval additivity
T (y)−T (x) = V y
x (f) ≥ |f(y)−f(x)| ≥ ±(f(y)−f(x)) if a ≤ x ≤ y ≤ b
it also follows that T is non-decreasing, as are P = 1(T
+ f − f(a))
2
and N = 1(T
− f + f(a)). But then f = (f(a) + P ) − N is a splitting
2
of f into a difference of non-decreasing functions. Note also that T =
P + N. Conversely, if u and v are non-decreasing functions on [a, b]