Riemann–Stieltjes Integral

Definition

For a bounded function $f$, a partition $P$ of a non-decreasing function $g$ on $[a,b]$,

Let $f$ be a bounded function, $P$ be a partition of a non-decreasing function $g$ on a $[a,b]$, $\Delta g_i=g(x_i)-g(x_{i-1})$ $M_i=\sup \{f(x)|x_{i-1}\le x\le x_i\}$, and $m_i=\inf \{f(x)|x_{i-1}\le x\le x_i\}$

Stieltjes Sums

The upper Stieltjes sum of $f$ with respect to $P$ is $U_{f,P,g}=\sum _{i=1}^n{M}_i\Delta g_i$

The lower Stieltjes sum of $f$ with respect to $P$ is $L_{f,P,g}=\sum _{i=1}^n{m}_i\Delta g_i$

Stieltjes Integrals

The upper Stieltjes integral of $f$ is $U_{f,g}=\overline{{\int }_a^b}f(x)dg(x)=\inf \{U_{f,P,g}\}$

The lower Stieltjes integral of $f$ is $L_{f,g}=\underline{{\int }_a^b}f(x)dg(x)=\sup \{L_{f,P,g}\}$

If $U_{f,g}=L_{f,g}\iff \overline{{\int }_a^b}f(x)dg(x)=\underline{{\int }_a^b}f(x)dg(x)$, then we call the common value the Riemann–Stieltjes integral and set ${\int }_a^{b}f(x)dg(x)=U_{f,g}=L_{f,g}$

We also say that $f$ is Riemann–Stieltjes-integrable or $f\in {\mathcal{R}}_g[a,b]$, where ${\mathcal{R}}_g[a,b]$ is a set of Riemann–Stieltjes integrable function on a $[a,b]$

Facts

Suppose $f\in \mathcal{R}[a,b]$, $g$ is non-decreasing differentiable function, $g^{\prime }\in {\mathcal{R}}_g[a,b]$, then $f\in {\mathcal{R}}_g[a,b]$ and the following is satisfied ${\int }_a^{b}f(x)dg(x)={\int }_a^{b}f(x)g^{\prime }(x)dx$