The Additivity Over Domains of Int. Prop. of Nonneg. Meas. Functs.

# The Additivity Over Domains of Integration Property of Nonnegative Measurable Functions

 Theorem 1 (The Finite Additivity Over Domains of Integration): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ be a nonnegative measurable function defined on the measurable sets $A$ and $B$ where $A$ and $B$ are mutually disjoint. Then $\displaystyle{\int_{A \cup B} f(x) \: d \mu = \int_A f(x) \: d \mu + \int_B f(x) \: d \mu}$.
• Proof: Let $f$ be a nonnegative measurable function defined on the measurable sets $A$ and $B$.
(1)
\begin{align} \quad \lim_{n \to \infty} \int_{A \cup B} [\varphi_n(x) + \varphi_n'(x)] \: d \mu &= \int_{A \cup B} f(x) \: d \mu \\ \quad \lim_{n \to \infty} \int_{A \cup B} \varphi_n(x) \: d \mu + \lim_{n \to \infty} \int_{A \cup B} \varphi_n'(x) \: d \mu &= \int_{A \cup B} f(x) \: d \mu \\ \quad \lim_{n \to \infty} \int_A \varphi_n \: d \mu + \lim_{n \to\infty} \int_B \varphi_n'(x) \: d \mu &= \int_{A \cup B} f(x) \: d \mu \\ \quad \int_A f(x) \: d \mu + \int_B f(x) \: d \mu &= \int_{A \cup B} f(x) \: d \mu \quad \blacksquare \end{align}
 Theorem 2 (The Countable Additivity Over Domains of Integration): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ be a nonnegative measurable function defined on the measurable sets $(E_n)_{n=1}^{\infty}$ where $(E_n)_{n=1}^{\infty}$ is a collection of mutually disjoint sets. Then $\displaystyle{\int_{\bigcup_{n=1}^{\infty} E_n} f(x) \: d \mu = \sum_{n=1}^{\infty} \int_{E_n} f(x) \: d \mu}$.