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393
O46
ž¦·¡´œ›r¦¸¤´œœr-˜¸¨˜rÁ
On Riemann-Stieltjes Integral
ª¸¦³´¥ šn µ—¸1 ª¦µ¥» »ž„‡¦°Š 2 ¨³‹´œšœµ Å°¥¦µ„µ‹œ„»¨3
Weerachai Thadee, Varayu Boonpogkrong and Jantana Ayaragarnchanakul
š‡´—¥n °
ž¦·¡´œ›r¦¸¤´œœr-˜¸¨˜rÁ ™¼ „œ·¥µ¤ÅªoĜÁ¦¸¥œ‡–· ˜ª·Á‡¦µ³®rµŠÁ¨n¤ Kenneth A. Ross ŗo—Š
„µ¦¡· ¼‹œrĜÁ¦¸¥œ Elementary Analysis: The Theory of Calculus Ūoªnµ ™o µ˜´ª®µž¦·¡´ œ›rÁž}œ¢{Š„r´œ˜n°ÁœºÉ °Š
¨³Á‡¦ºÉ °Š®µž¦·¡´ œ›rÁž}œ¢{Š„r´œÁ¡·É¤ ¨oªž¦· ¡´œ›r¦¸¤´œœr-˜¸¨˜rÁ‹³¤¸°¥¼n‹¦·Š
Ĝ„µ¦«¹„¬µ‡¦´ÊŠœ¸Ê Á¦µ‹³¨—ÁŠºÉ°œÅ „µ¦¤¸°¥¼n ‹¦·Š °Šž¦· ¡´œ›r¦¸¤´œœr-˜¸¨˜rÁ ×¥„µ¦¡· ¼‹œrªnµ ™oµ
¢{ Š„r´œ [ , ] f RF a b ¨³¢{Š„r´œ [ , ] g BV a b ¨oª ( )
b
a
RS fdg³ ‹³¤¸°¥¼n‹¦·Š
‡ÎµÎµ‡´ : ž¦·¡´œ›r¦¸¤´œœr-˜¸¨˜rÁ ¢{ Š„r´œ regulated „µ¦Âž¦Ÿ´œÂ¤¸°Á ˜
Abstract
The Riemann-Stieltjes integral is defined in some text books on analysis. Kenneth A. Ross proved in his
Elementary Analysis: The Theory of Calculus, that if the integrand is a continuous function and integrator is an
increasing function then the Riemann-Stieltjes integral exists.
In this study, we shall weaken the condition on the Riemann-Stieltjes integral. More precisely, we prove
that if [ , ] f RF a b and [ , ] g BV a b , then ( )
b
a
RS fdg³ exists.
Keywords: Riemann-Stieltjes integral, Regulated function, Bounded variation
1œ´„«¹ „¬µž¦·µÃš £µ‡ª·µ‡–·˜«µ˜¦r ¤®µª·š¥µ¨´¥Š ¨µœ‡¦· œš¦r Š ¨µ 90112
Master's Degree student, Department of Mathematics, Faculty of Science, Prince of Songkla University, Songkhla, 90112.
Corresponding author : Ú¦«´¡šr 08 4993 1275 E-mail :weerachai6075@gmail.com
2—¦. £µ‡ª·µ‡–· ˜«µ˜¦r ‡–³ª·š¥µ«µ˜¦r ¤®µª·š¥µ¨´¥Š ¨µœ‡¦· œš¦r Š ¨µ 90112
Dr., Department of Mathematics, Faculty of Science, Prince of Songkla University, Songkhla, 90112
3 ¦«.—¦. £µ‡ª·µ‡–·˜«µ˜¦r ‡–³ª·š¥µ«µ˜¦r ¤®µª· š¥µ¨´¥Š ¨µœ‡¦· œš¦r Š ¨µ 90112
Assoc. Prof. Dr., Department of Mathematics, Faculty of Science, Prince of Songkla University, Songkhla, 90112