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<x_1<\dots<x_n=b\}, \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].


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