A Study of a Stieltjes Integral Defined on Arbitrary Number Sets

Abstract

Our purpose is to study a generalized Stieltjes integral defined on a class of subsets of a closed number interval. We extend the results of previous work by the first author. Among other results, we prove that

If $M \subseteq [a,b]$ and $f$ and $g$ are functions with domain $M$ such that $f$ is $g$-integrable over $M$, and there exist left (right) extensions $f^*$ and $g^*$ of $f$ and $g$ to $[a,b]$, respectively, then $f^*$ is $g^*$- integrable on $[a,b]$ and $$ \int_a^b f^*dg^*= \int_M fdg $$

[(a)] $F$ is $G$-integrable on $[a,b]$,

[(b)] $\overline{M} \subseteq [a,b]$, and $a,b \in M$ \item

[(c)] if $z$ belongs to $[a,b] - M$ and $\epsilon$ is a positive number, then there is an open interval $s$ containing $z$ such that \break $|F(x) - F(z)||G(v) - G(u)| <\epsilon$ where each of $u$, $v$, and $x$ is in $s \cap [a,b]$, $u < z < v$, and $u \le x \le v$.

Then $F$ is $G$-integrable on $M$, and $\int_a^b FdG = \int\limits_{M}FdG$.