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 $\displaystyle{\int_E (\alpha f(x) + \beta g(x)) \: d \mu = \alpha \int_E f(x) \: d \mu + \beta \int_E g(x) \: d \mu}$. |
- Proof: Let $f$ and $g$ be nonnegative measurable functions defined on a measurable set $E$. By The Simple Function Approximation Lemma and Theorem for General Measurable Spaces there exists pointwise increasing sequences $(\varphi_n(x))_{n=1}^{\infty}$ and $(\psi_n(x))_{n=1}^{\infty}$ of nonnegative measurable functions that converge pointwise to $f(x)$ and $g(x)$ respectively.
- Let $\alpha, \beta \in \mathbb{R}$. Then $(\alpha \varphi_n(x) + \beta \psi_n(x))_{n=1}^{\infty}$ is a pointwise increasing sequence of nonnegative measurable functions that converge pointwise to $\alpha f(x) + \beta f(x)$ on $E$. So by The Monotone Convergence Theorem for Nonnegative Measurable Functions we have that:
\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}