Riemann-Stieltjes integral

Let $f$ and $\alpha$ be bounded, real-valued functions defined upon a closed finite interval $I=[a,b]$ of $ℝ(a\ne b)$, $P=\{x_0,\mathrm{\dots },x_n\}$ a partition of $I$, and $t_i$ a point of the subinterval $[x_{i-1},x_i]$. A sum of the form

$$S(P,f,\alpha )=\sum _{i=1}^{n}f(t_i)(\alpha (x_i)-\alpha (x_{i-1}))$$

is called a Riemann-Stieltjes sum of $f$ with respect to $\alpha$. $f$ is said to be Riemann Stieltjes integrable with respect to $\alpha$ on $I$ if there exists $A\in ℝ$ such that given any $ϵ>0$ there exists a partition $P_{ϵ}$ of $I$ for which, for all $P$ finer than $P_{ϵ}$ and for every choice of points $t_i$, we have

If such an $A$ exists, then it is unique and is known as the Riemann-Stieltjes integral of $f$ with respect to $\alpha$. $f$ is known as the integrand and $\alpha$ the integrator. The integral is denoted by

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