Riemann–Stieltjes integral

An integral ∫ f dα defined via limits of sums using increments of an integrator α.

Riemann–Stieltjes integral

Let $f:[a,b]\to\mathbb{R}$ be bounded, and let $\alpha:[a,b]\to\mathbb{R}$ be a function. For a partition $P:a=x_0<\cdots<x_n=b$ , define

\Delta\alpha_i:=\alpha(x_i)-\alpha(x_{i-1}).

For each $i$ , set

M_i:=\sup_{x\in[x_{i-1},x_i]} f(x),\qquad m_i:=\inf_{x\in[x_{i-1},x_i]} f(x).

The upper and lower Riemann–Stieltjes sums are

U(f,\alpha;P):=\sum_{i=1}^n M_i\,\Delta\alpha_i,\qquad L(f,\alpha;P):=\sum_{i=1}^n m_i\,\Delta\alpha_i.

If $\alpha$ is increasing, one says $f$ is Riemann–Stieltjes integrable with respect to $\alpha$ if

\sup_P L(f,\alpha;P)=\inf_P U(f,\alpha;P),

and the common value is denoted

\int_a^b f\,d\alpha.

The Riemann–Stieltjes integral generalizes the Riemann integral: taking $\alpha(x)=x$ recovers $\int_a^b f(x)\,dx$ . It also encodes integration with respect to step functions (leading to sums) and is the classical precursor to Lebesgue–Stieltjes integration.

Examples:

If $\alpha(x)=x$ , then $\int_a^b f\,d\alpha=\int_a^b f(x)\,dx$ (Riemann integral).
If $\alpha$ is constant, $\alpha(x)\equiv c$ , then $\Delta\alpha_i=0$ for all $i$ and $\int_a^b f\,d\alpha=0$ .
If $\alpha$ is a step function with a jump at $c\in(a,b)$ , then $\int_a^b f\,d\alpha$ reduces (under standard integrability assumptions) to a multiple of $f(c)$ reflecting the jump size.