Spectral Theory in Hilbert Space

B. STIELTJES INTEGRALS 125
and {x0, . . . , xn} a partition of [a, x], a < x ≤ b, then
n
|(u(xk) − v(xk)) − (u(xk−1) − v(xk−1))|
k=1
≤
n
|u(xk) − u(xk−1)| +
k=1
n
|v(xk) − v(xk−1)|
k=1
= u(x) − u(a) + v(x) − v(a),
so that V x
a (u−v) ≤ u(x)+v(x)−(u(a)+v(a)). In particular, for x = b
this shows that u − v is of bounded variation on [a, b]. The inequality
also shows that if f = u − v, then
P (x) = 1(T
(x) + f(x) − f(a))
2
≤ 1
2
(u(x) − u(a) + v(x) − v(a) + f(x) − f(a)) = u(x) − u(a) .
Similarly one shows that N(x) ≤ v(x) − v(a) so that the proof is
complete.
We remark that a complex-valued function (of a real variable) is
said to be of bounded variation if its real and imaginary parts are. If
Tr and Ti are the total variation functions of the real and imaginary
parts of f, then one defines the total variation function of f to be
T = T 2 r + T 2
i (sometimes the definition T = Tr + Ti is used). One
may also use Definition B.5 for complex-valued functions, and then it
is easily seen that T 2 r + T 2
i ≤ T ≤ Tr + Ti.
Since a monotone function can have only jump discontinuities, and
at most countably many of them, also functions of bounded variation
can have at most countably many discontinuities, all of them jump discontinuities.
Moreover, it is easy to see that the positive and negative
variation functions (and therefore the total variation function) are continuous
wherever f is (Exercise B.7).
Corollary B.7. If g is of bounded variation on [a, b], then every
continuous function f is integrable with respect to g and we have
b
(B.2)
f dg ≤ max|f|V
[a,b]
b
a (g).
a
Proof. The integrability statement follows immediately from Theorem
B.4 on writing g as the difference of non-decreasing functions. To
obtain the inequality, consider a Riemann-Stieltjes sum
s =
n
f(ξk)(g(xk) − g(xk−1)).
k=1