Spectral Theory in Hilbert Space

B. STIELTJES INTEGRALS 123
Note that Theorem B.3 is a statement about the Riemann-Stieltjes
integral; for more general (Lebesgue-Stieltjes) integrals it is not true
without further assumptions about f and g. The reason is that the
Riemann-Stieltjes integrals can not exist if f and g have discontinuities
in common (Exercise B.4), whereas the Lebesgue-Stieltjes integrals exist
as soon as f and g are, for example, both monotone. In such a case
the integration by parts formula only holds under additional assumptions,
for example if f is continuous to the right and g to the left in any
common point of discontinuity, or if both f and g are normal, i.e., their
values at points of discontinuity are the averages of the corresponding
left and right hand limits.
So far we don’t know that any function is integrable with respect
to any other (except for g(x) = x which is the case of the Riemann
integral).
Theorem B.4. If g is non-decreasing on [a, b], then every continuous
function f is integrable with respect to g and we have
b
f dg ≤ max|f|(g(b)
− g(a)).
[a,b]
a
Proof. Let ∆ ′ and ∆ ′′ be partitions a = x ′ 0 < x ′ 1 < · · · < x ′ m = b
and a = x ′′
0 < x ′′
1 < · · · < x ′′ n = b of [a, b] and consider the corresponding
Riemann-Stieltjes sums s ′ = m
k=1 f(ξ′ k )(g(x′ k ) − g(x′ k−1 )) and s′′ =
n
k=1 f(ξ′′
k )(g(x′′
k )−g(x′′
k−1 )). If we introduce the partition ∆ = ∆′ ∪∆ ′′ ,
supposing it to be a = x0 < x1 < · · · < xp = b, we can write
s ′ − s ′′ p
= (f(ξ ′ kj ) − f(ξ′′ qj ))(g(xj) − g(xj−1))
j=1
where kj = k for all j for which [xj−1, xj] ⊂ [x ′ k−1 , x′ k ] and qj = k for
all j for which [xj−1, xj] ⊂ [x ′′
k−1 , x′′ k ] (check this carefully!). Thus, for
all j, ξ ′ kj and xj are in the same subinterval of the partition ∆ ′ , and
ξ ′′
qj and xj in the same subinterval of the partition ∆ ′′ . It follows that
|ξ ′ − ξ′′ kj qj | ≤ |ξ′ kj − xj| + |ξ ′′
qj − xj| ≤ |∆ ′ | + |∆ ′′ | for all j. Since f
is uniformly continuous on [a, b], this means that given ε > 0, then
|f(ξ ′ ) − f(ξ′′
kj qj )| ≤ ε if |∆′ | and |∆ ′′ | are both small enough. It follows
that |s ′ −s ′′ | ≤ ε p j=1 |g(xj)−g(xj−1| = ε(g(b)−g(a)) for small enough
|∆ ′ | and |∆ ′′ |. Thus f is integrable with respect to g according to
Proposition B.1. We also have |s ′ | ≤ n k=1 |f(ξ′ k )||g(x′ k ) − g(x′ k−1 )| ≤
max|f|(g(b) − g(a)) so the proof is complete.
As a generalization of Theorem B.4 we may of course take g to be
any function which is the difference of two non-decreasing functions.
Such a function is called a function of bounded variation. We shall
briefly discuss such functions; the main point is that they are characterized
by having finite total variation.