Spectral Theory in Hilbert Space

APPENDIX B
Stieltjes integrals
The Riemann-Stieltjes integral is a simple generalization of the
(one-dimensional) Riemann integral. To define it, let f and g be two
functions defined on the compact interval [a, b]. For every partition
∆ = {xj} n j=0 of [a, b], i.e., a = x0 < x1 < · · · < xn = b, we let the mesh
of ∆ be |∆| = max(xk − xk−1). This is the length of the longest subinterval
of [a, b] in the partition. We also choose from each subinterval
[xk−1, xk] a point ξk and form the sum
s =
n
f(ξk)(g(xk) − g(xk−1)) .
k=1
Now suppose that s tends to a limit as |∆| → 0 independently of the
partition ∆ and choice of the points ξk. The exact meaning of this is
the following: There exists a number I such that for every ε > 0 there
is a δ > 0 such that |s − I| < ε as soon as |∆| < δ. In this case we say
that the integrand f is Riemann-Stieltjes integrable with respect to the
integrator g and that the corresponding integral equals I. We denote
this integral by b
a f(x) dg(x) or simply b
f dg. The choice g(x) = x
a
gives us, of course, the ordinary Riemann integral.
Proposition B.1. A function f is integrable with respect to a function
g if and only if for every ε > 0 there exists a δ > 0 such that for
any two partitions ∆ and ∆ ′ and the corresponding sums s and s ′ , we
have |s − s ′ | < ε as soon as |∆| and |∆ ′ | are both < δ.
This is of course a version of the Cauchy convergence principle. We
leave the proof as an exercise (Exercise B.1). From the definition the
following calculation rules follow immediately (Exercise B.2).
(1)
b
a
(2) C
(3)
b
a
f1 dg +
b
a
b
a
f dg =
f dg1 +
a
b
a
b
f2 dg =
Cf dg,
f dg2 =
b
a
b
a
(f1 + f2) dg,
f d(g1 + g2),
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