Integral of Positive Measurable Function is Additive

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Theorem

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

Let $f : X \to \overline \R$ and $g : X \to \overline \R$ be positive $\Sigma$-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 additive.


Corollary

Let $A \in \Sigma$.


Then:

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

where:

$f + g$ is the pointwise sum of $f$ and $g$
the integral sign denotes $\mu$-integration over $A$.


This can be summarized by saying that $\ds \int_A \cdot \rd \mu$ is additive.


Proof

We are given that $f : X \to \overline \R$ and $g : X \to \overline \R$ is a positive $\Sigma$-measurable functions, which is a fortiori a measurable function, so we can apply Measurable Function is Pointwise Limit of Simple Functions.

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 the Sum Rule for Real Sequences, 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:

\(\ds \int \paren {f + g} \rd \mu\) \(=\) \(\ds \lim_{n \mathop \to \infty} \int \paren {f_n + g_n} \rd \mu\) Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren {\int f_n \rd \mu + \int g_n \rd \mu}\) Integral of Positive Simple Function is Additive
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \int f_n \rd \mu + \lim_{n \mathop \to \infty} \int g_n \rd \mu\) Sum Rule for Real Sequences
\(\ds \) \(=\) \(\ds \int f \rd \mu + \int g \rd \mu\) Integral of Positive Measurable Function as Limit of Integrals of Positive Simple Functions

$\blacksquare$


Sources

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