Integral of Positive Measurable Function is Additive

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f, g: X \to \overline \R$, $f, g \in \MM_{\overline \R}^+$ be positive measurable functions.

Then:


 * $\ds \int \paren {f + g} \rd \mu = \int f \rd \mu + \int g \rd \mu$

where:
 * $f + g$ is the pointwise sum of $f$ and $g$
 * the integral sign denotes $\mu$-integration

This can be summarized by saying that $\ds \int \cdot \rd \mu$ is (conventionally) additive.

Proof
From Measurable Function is Pointwise Limit of Simple Functions, there exists an increasing sequence $\sequence {f_n}_{n \mathop \in \N}$ of positive simple functions such that:


 * $\ds \map f x = \lim_{n \mathop \to \infty} \map {f_n} x$

for each $x \in X$.

Similarly, there exists an increasing sequence $\sequence {g_n}_{n \mathop \in \N}$ of positive simple functions such that:


 * $\ds \map g x = \lim_{n \mathop \to \infty} \map {g_n} x$

for each $x \in X$.

From Combination Theorem for Sequences: Real: Sum Rule, we have:


 * $\ds \map f x + \map g x = \lim_{n \mathop \to \infty} \paren {\map {f_n} x + \map {g_n} x}$

for each $x \in X$.

We then have: