RiemannStieltjes integral
Let $f$ and $\alpha $ be bounded^{}, realvalued functions defined upon a closed finite interval $I=[a,b]$ of $\mathbb{R}(a\ne b)$, $P=\{{x}_{0},\mathrm{\dots},{x}_{n}\}$ a partition^{} of $I$, and ${t}_{i}$ a point of the subinterval $[{x}_{i1},{x}_{i}]$. A sum of the form

$$S(P,f,\alpha )=\sum _{i=1}^{n}f({t}_{i})(\alpha ({x}_{i})\alpha ({x}_{i1}))$$ 

is called a RiemannStieltjes sum of $f$ with respect to $\alpha $. $f$ is said to be Riemann Stieltjes integrable with respect to $\alpha $ on $I$ if there exists $A\in \mathbb{R}$ such that given any $\u03f5>0$ there exists a partition ${P}_{\u03f5}$ of $I$ for which, for all $P$ finer than ${P}_{\u03f5}$ and for every choice of points ${t}_{i}$, we have
If such an $A$ exists, then it is unique and is known as the RiemannStieltjes integral of $f$ with respect to $\alpha $. $f$ is known as the integrand and $\alpha $ the integrator. The integral is denoted by

$${\int}_{a}^{b}f\mathit{d}\alpha \mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{\int}_{a}^{b}f(x)\mathit{d}\alpha (x)$$ 
