Infinite Measure is Measure
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):
- $\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}$
Suppose $S_n = \O$ for all $n \in \N$.
Then by definition of $\mu$:
- $\ds \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} = \map \mu \O = 0$
Thus, the desired equation becomes:
- $\ds \sum_{n \mathop = 1}^\infty 0 = 0$
which trivially holds.
Suppose $S_n \ne \O$ for some $n \in \N$.
Then $S_n \subseteq \ds \bigcup_{n \mathop = 1}^\infty S_n \ne \O$.
- $\ds \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} = + \infty$
- $\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} = \map \mu {S_n} = + \infty = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}$
Thus $\ds \sum_{n \mathop = 1}^\infty \map \mu {S_n} = \map \mu {\bigcup_{n \mathop = 1}^\infty S_n}$.
$\Box$
The axioms are fulfilled, and it follows that $\mu$ is a measure.
$\blacksquare$