TITLE: Question about the existence of Riemann-Stieltjes integrals.
QUESTION [0 upvotes]: I learned like the following:
For Riemann integrals,
$$
\int_a^b fd\phi \quad \text{exists} \quad \Leftrightarrow \quad \sup_\Gamma L_\Gamma = \inf_\Gamma L_\Gamma
$$
where $\Gamma$ is a partition of $[a, b]$, $$\Gamma=\left\{x_i\right\}_{i=0}^m \quad \text{satisfying} \quad x_0=a < x_1 < \cdots <x_m=b$$

For Riemann-Stieltjes integrals,
$$
\int_a^b fd\phi \quad \text{exists} \quad \nLeftrightarrow \quad \sup_\Gamma L_\Gamma = \inf_\Gamma L_\Gamma
$$
where $\Gamma$ is a partition of $[a, b]$, $$\Gamma=\left\{x_i\right\}_{i=0}^m \quad \text{satisfying} \quad x_0=a < x_1 < \cdots <x_m=b$$

My question is whether the following equation is true or not
For Riemann-Stieltjes integrals,
$$
\int_a^b fd\phi \quad \text{exists} \quad \Leftrightarrow \quad \lim_{|\Gamma|\to0}R_\Gamma\quad\text{exists}.
$$
where $R_\Gamma=\displaystyle\sum_{i=1}^{m}f(\xi_i)[\phi(x_i)-\phi(x_{i-1})]$, $\xi_i\in[x_{i-1}, x_i]$.

REPLY [1 votes]: If $\{ x_i\}$ is taken to be: "any partition of $[a,b]$ of fineness less than or equal to $|\Gamma|$" then you have just written the definition of the Riemann-Stieltjes integral, so the answer to your question is yes.
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