## Generalized Riemann-Stieltjes Integral

The generalized Riemann-Stieltjes integral $\int_a^b f\,d\phi$ is a number $\gamma$ such that: for every $\epsilon>0$ there exists a partition $P_\epsilon$ of $[a,b]$ such that if $\dot{P}=(P,\xi)$, $P=\{a=x_0, $\xi=\{\xi_i: i=1,\dots, n\}$ with $\xi_i\in[x_{i-1},x_i]$ is a tagged partition of $[a,b]$ such that $P$ is a refinement of $P_\epsilon$, then $\displaystyle |S(\dot{P},f,\phi)-\gamma|=|\sum_{i=1}^n f(\xi_i)(\phi(x_i)-\phi(x_{i-1}))-\gamma|<\epsilon.$

We will write $\displaystyle \lim_{P\to 0}S(\dot{P},f,\phi)=\lim_{P\to 0}\sum_{i=1}^nf(\xi_i)(\phi(x_i)-\phi(x_{i-1}))=\gamma$ and $f\in \mathcal{R}_\phi[a,b]$.