Linearity of the Riemann–Stieltjes integral

Riemann–Stieltjes linearity: Let $a<b$ and let $f,h,g_1,g_2:[a,b]\to\mathbb{R}$. Suppose the indicated Riemann–Stieltjes integrals exist. For scalars $\alpha,\beta\in\mathbb{R}$:

• If $\int_a^b f\,dg$ and $\int_a^b h\,dg$ exist, then $\int_a^b (\alpha f+\beta h)\,dg$ exists and

  $$\int_a^b (\alpha f+\beta h)\,dg = \alpha\int_a^b f\,dg+\beta\int_a^b h\,dg.$$

• If $\int_a^b f\,dg_1$ and $\int_a^b f\,dg_2$ exist, then $\int_a^b f\,d(\alpha g_1+\beta g_2)$ exists and

  $$\int_a^b f\,d(\alpha g_1+\beta g_2) = \alpha\int_a^b f\,dg_1+\beta\int_a^b f\,dg_2.$$

These identities are the Riemann–Stieltjes analog of linearity of the Riemann integral and are used routinely together with integration by parts .