Chapter 1
The Riemann-Stieltje s
Integral
1.1. Propertie s o f th e Riemann-Stieltje s Integra l
We star t wit h som e basi c notations , definition s an d theorems . B y a
partition P o f a close d interva l [a , b] we mea n a finite se t o f point s
xo, xi,..., xn suc h tha t
a =
XQ
x\ .. . x
n
-\ x
n
= b.
The numbe r n(P) = maxjx ^ xi-\ : i = 1, 2 , . .. ,n} i s calle d th e
raes/i of P.
For a functio n a monotonicall y increasin g o n [a , b] w e writ e
A ^ = a(xi) - a(xi-i).
If / i s a real function bounde d o n [a , 6], we define th e uppe r an d lowe r
Darboux sum s o f / wit h respec t t o a an d relativ e t o P , respectively ,
by
n n
U(P,f,a) = Y,MiAa
t
, L(P,f,a) = ^ m ^ A a , ,
where
Mi = su p /(#) , ^ i = in f /(#)
3
http://dx.doi.org/10.1090/stml/021/01
Previous Page Next Page