Countably Additive Function also Finitely Additive

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Theorem

Let $\mathcal A$ be a $\sigma$-algebra.

Let $f: \mathcal A \to \overline {\R}$ be a function, where $\overline {\R}$ denotes the set of extended real numbers.

$\displaystyle f \left({\bigcup_{i \in \N} A_i}\right) = \sum_{i \in \N} f \left({A_i}\right)$

such that there exists at least one $A \in \mathcal A$ where $f \left({A}\right)$ is a finite number.

We have that $f$ is defined as countably additive iff:

$\displaystyle f \left({\bigcup_{i \ge 1} A_i}\right) = \sum_{i \ge 1} f \left({A_i}\right)$

where $\left \langle {A_i} \right \rangle$ is any sequence of pairwise disjoint elements of $\mathcal A$.

We need to show that:

$\displaystyle \forall n \in \N: f \left({\bigcup_{i=1}^n A_i}\right) = \sum_{i=1}^n f \left({A_i}\right)$

Let $n \in \N$ be any arbitrary natural number.

Let $\left \langle {B_i}\right \rangle$ be the sequence of pairwise disjoint elements of $\mathcal A$ defined as:

$B_i = \begin{cases} A_i & : i \le n \\ \varnothing & : i > n \end{cases}$

It follows that:

$\displaystyle \bigcup_{i \ge 1} B_i = \bigcup_{i = 1}^n A_i$

Thus

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \mu \left({\bigcup_{i = 1}^n A_i}\right)$	$=$	$\displaystyle \mu \left({\bigcup_{i \ge 1} B_i}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sum_{i \ge 1} f \left({B_i}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sum_{i=1}^n f \left({A_i}\right) + \sum_{i>n} f \left({\varnothing}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sum_{i=1}^n f \left({A_i}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from Countably Additive Function of Null Set

Hence the result.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mu \left({\bigcup_{i = 1}^n A_i}\right)\)	\(=\)	\(\displaystyle \mu \left({\bigcup_{i \ge 1} B_i}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{i \ge 1} f \left({B_i}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{i=1}^n f \left({A_i}\right) + \sum_{i>n} f \left({\varnothing}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{i=1}^n f \left({A_i}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from Countably Additive Function of Null Set