Definition:Riemann-Stieltjes Integral/General Limits of Integration

Definition

The notation for the Riemann-Stieltjes integral:

$\ds A = \int_a^b f \rd \alpha = \int_a^b \map f x \rd \map \alpha x$

is defined for $a < b$.

We extend it to arbitrary $a, b \in \R$ by the following conventions.

If $a = b$, then:

$\ds \int_a^b f \rd \alpha = \int_a^b \map f x \rd \map \alpha x := 0$

If $a > b$, then:

$\ds \int_a^b f \rd \alpha = \int_a^b \map f x \rd \map \alpha x := - \int_b^a f \rd \alpha$

whenever $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint b a$.

Sources

• 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $7$ The Riemann-Stieltjes Integral: $\S 7.4$: Definition $7.5$