Infinite Measure is Measure

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Theorem

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


Then the infinite measure $\mu$ on $\struct {X, \Sigma}$ is a measure.


Proof

Let us verify the measure axioms $(1)$, $(2)$ and $(3')$ for $\mu$.


Proof of $(3')$

We have by definition $\map \mu \O = 0$.

$\Box$


Proof of $(1)$

Let $S \in \Sigma$.

Then:

$\map \mu S = + \infty > 0$ for $S \ne \O$
$\map \mu S = 0 \ge 0$ for $S = \O$

So $\map \mu S \ge 0$ for all $S \in \Sigma$.

$\Box$


Proof of $(2)$

It is to be shown that (for a sequence $\sequence {S_n}_{n \in \N}$ of pairwise disjoint sets):

$\displaystyle \sum_{n \mathop = 1}^\infty \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}$


Suppose $S_n = \O$ for all $n \in \N$.

Then by definition of $\mu$:

$\displaystyle \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} = \map \mu \O = 0$


Thus, the desired equation becomes:

$\displaystyle \sum_{n \mathop = 1}^\infty 0 = 0$

which trivially holds.


Suppose $S_n \ne \O$ for some $n \in \N$.

Then $S_n \subseteq \displaystyle \bigcup_{n \mathop = 1}^\infty S_n \ne \O$.

$\displaystyle \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} = + \infty$
$\displaystyle \sum_{n \mathop = 1}^\infty \map \mu {S_n} = \map \mu {S_n} = + \infty = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}$


Thus $\displaystyle \sum_{n \mathop = 1}^\infty \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}$.

$\Box$


The axioms are fulfilled, and it follows that $\mu$ is a measure.

$\blacksquare$