Let $f$ and $\alpha$ be bounded, real-valued functions defined upon a closed finite interval $I = [ a, b ]$ of $\mathbb{R} (a \neq b)$ , $P = \{ x_{0}, ..., x_{n} \}$ 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 $A \in \mathbb{R}$ such that given any $\epsilon > 0$ there exists a partition $P_{\epsilon}$ of $I$ for which, for all $P$ finer than $P_{\epsilon}$ and for every choice of points $t_{i}$ , we have

$$|S(P, f, \alpha) - A| < \epsilon$$

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

$$\int_{a}^{b}fd\alpha \quad \textrm{or} \quad \int_{a}^{b}f(x)d\alpha(x)$$