Real analysis/Riemann-Stieltjes integration

Definition:

Let {x_n} be a partition of of an interval [a,b], and let t_k be an element of [x_k-1,x_k], let f(x) be a function defined on [a,b], and let α(x) also be a function on [a,b]. If there exists a number L such that for every ε>0, there exists a partition {a_n} such that for any finer partition {x_n},

$|\sum _{k=1}^{n}f(t_k)(\alpha (x_k)-\alpha (x_{k-1}))-L|<ϵ$.

If this number L exists, then the function is called Riemann-Stieltjes integrable on the interval [a,b] and is denoted

$\underset{a}{\overset{b}{\int }}f(x)d\alpha$.