The Linearity Property of the Integral of Nonnegative Meas. Functions

# The Linearity Property of the Integral of Nonnegative Measurable Functions

 Theorem 1 (Linearity of the Integral of Nonnegative Measurable Functions): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ and $g$ be nonnegative measurable functions defined on a measurable set $E$. Then for all $\alpha, beta \in \mathbb{R}$ with $\alpha, \beta \geq 0$ we have that $\displaystyle{\int_E (\alpha f(x) + \beta g(x)) \: d \mu = \alpha \int_E f(x) \: d \mu + \beta \int_E g(x) \: d \mu}$.
(1)
\begin{align} \quad \lim_{n \to \infty} \int_E [\alpha \varphi_n(x) + \beta \psi_n(x)] \: d \mu &= \int_E [\alpha f(x) + \beta g(x)] \: d \mu \\ \quad \lim_{n \to \infty} \int_E \alpha \varphi_n(x) \: d \mu + \lim_{n \to \infty} \int_E \beta \psi_n(x) \: d \mu &= \int_E [\alpha f(x) + \beta g(x)] \: d \mu \\ \quad \alpha \lim_{n \to \infty} \int_E \varphi_n(x) \: d \mu + \beta \lim_{n \to \infty} \int_E \psi_n(x) \: d \mu &= \int_E [\alpha f(x) + \beta g(x)] \: d \mu \\ \quad \alpha \int_E f(x) \: d \mu + \beta \int_E g(x) \: d \mu &= \int_E [\alpha f(x) + \beta g(x)] \: d \mu \quad \blacksquare \end{align}