Usually, analysis textbooks do not classify Riemann-Stieltjes and Darboux-Stieltjes integral. But I know that a function is $\alpha$-DS integrable does not implies it is $\alpha$-RS integrable. Can anybody give me an example of that?

Definition of $\alpha$-DS integrable:

Let $P$ be a partition of $[a,b]$.

$M_i=\sup(f(x):x\in[x_i,x_{i-1}])$, $m_i=\inf(f(x):x\in[x_i,x_{i-1}])$


$U(P,f,\alpha)=\sum M_i\Delta \alpha_i$, $L(P,f,\alpha)=\sum m_i\Delta \alpha_i$

A function $f:[a,b]\to R$ is $\alpha$-DS integrable iff $\sup(P,f,\alpha)=\inf(P,f,\alpha)$ where the sup and inf are taken over all partition of $[a,b]$.

Definition of $\alpha$-RS integrable:

A function $f:[a,b]\to R$ is $\alpha$-RS integrable iff for any $\epsilon>0$, there exists $\delta>0$ such that for any partition $P$ with $mesh(P)<\delta$, $$\left|\sum f(\xi_i)\Delta \alpha-\int^b_afd\alpha\right|<\epsilon,\quad \forall\xi_i\in[x_i,x_i-1]$$

  • $\begingroup$ Generally speaking, $RS$ is stronger than $DS.$ And when $\alpha$ is monotonous, these two definitions are equivalent. $\endgroup$ – painday Jan 25 '18 at 9:13

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