Additivity and linearity lemmas for Riemann and Riemann–Stieltjes integrals

Additivity and linearity (Riemann integral): Let $f,g:[a,b]\to\mathbb{R}$ be Riemann integrable and let $\alpha,\beta\in\mathbb{R}$. Then:

• $\alpha f+\beta g$ is Riemann integrable on $[a,b]$, and $\int_a^b (\alpha f(x)+\beta g(x))\,dx = \alpha\int_a^b f(x)\,dx+\beta\int_a^b g(x)\,dx.$
• For any $c\in[a,b]$, $f$ is Riemann integrable on $[a,c]$ and on $[c,b]$, and $\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.$

Additivity and linearity (Riemann–Stieltjes integral): Let $\gamma:[a,b]\to\mathbb{R}$ be increasing. If $f,g$ are Riemann–Stieltjes integrable with respect to $\gamma$ on $[a,b]$ and $\alpha,\beta\in\mathbb{R}$, then $\alpha f+\beta g$ is Riemann–Stieltjes integrable with respect to $\gamma$ and $\int_a^b (\alpha f+\beta g)\,d\gamma = \alpha\int_a^b f\,d\gamma+\beta\int_a^b g\,d\gamma.$ Moreover, for any $c\in[a,b]$, $\int_a^b f\,d\gamma=\int_a^c f\,d\gamma+\int_c^b f\,d\gamma.$