Linearity and Monotonicity of the Lebesgue Integral

Let $(X,\mathcal{F},\mu)$ be a measure space.

1. (Nonnegative case.) Let $f,g:X\to[0,\infty]$ be measurable and let $c\in[0,\infty)$. Then $f+g$ and $cf$ are measurable, and

with the conventions of Measure, Measure Space, and Probability Measure; and if $f(x)\le g(x)$ for all $x$ then $\int_X f\,d\mu\le\int_X g\,d\mu$.

2. (Integrable case.) Let $f,g:X\to\mathbb{R}$ be integrable and let $a,b\in\mathbb{R}$. Then $af+bg$ is integrable and

Moreover $\bigl|\int_X f\,d\mu\bigr|\le\int_X|f|\,d\mu$, and if $f(x)\le g(x)$ for all $x$ then $\int_X f\,d\mu\le\int_X g\,d\mu$.