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. $

These are the basic algebraic rules that make integration behave like a linear functional and allow interval decomposition.

Proof sketch: Linearity follows because the corresponding Riemann sums (or Riemann–Stieltjes sums) are linear in the integrand, and the upper /lower sums satisfy compatible inequalities. Additivity over $[a,c]$ and $[c,b]$ follows by splitting any partition of $[a,b]$ at the point $c$ and observing that sums decompose as a sum over the two subintervals.