398
Therefore, by Cauchy’s criterion for Darboux-Stieltjes integral,
f
is
DS
-integrable on
[ , ]
a b
with respect to
g
.
Theorem 4.2.
If
[ , ]
f RF a b
and
g
is an increasing function, then
f
is
RS
-integrable on
[ , ]
a b
with respect
to
g
.
Proof
. By Theorems 1.9 and 4.1, we get the required result.
Theorem 4.3.
[ , ]
f RF a b
and
[ , ]
g BV a b
, then
f
is
RS
-integrable on
[ , ]
a b
with respect to
g
.
Proof
. Let
[ , ]
f RF a b
and
[ , ]
g BV a b
. By Theorem 2.5,
g
is the difference of two increasing
functions. By Theorems 1.7 and 4.2,
f
is
RS
-integrable on
[ , ]
a b
with respect to
g
.
References
K. K. Aye. (2002).
The Duals of Some Banach Spaces
, Ph.D. Thesis, National Institute of Education, Singapore.
Varayu Boonpogkrong and Chew Tuan Seng. (2004/2005).
On integrals with integrators in BV
p
, Real Analysis
Exchange, 30(1), 193-200.
Ross, Kenneth A. (1980).
Elementary Analysis: The Theory of Calculus
, New-York: Springer-Verlag.
L. C. Young. (1936).
inequality of the Holder type, connected with Stieltjes integration
, Acta Math., 67,
251-282.
L. C. Young. (1938)
. General inequalities for Stieltjes integrals and the convergence of Fourier series
,
Mathematische Annalen, 115, 581-612.
