Definition:Co-Countable Measure

Theorem
Let $X$ be an uncountable set.

Let $\Sigma$ be the $\sigma$-algebra of countable sets on $X$.

Then the co-countable measure (on $X$) is the measure $\mu: \Sigma \to \overline \R$ defined as:


 * $\forall E \in \Sigma: \map \mu E := \begin{cases} 0 & : \text {if $E$ is countable} \\ 1 & : \text {if $E$ is co-countable}\end{cases}$

where:
 * $\overline \R$ denotes the extended real numbers
 * $E$ is co-countable $X \setminus E$ is countable.

Also see

 * Co-Countable Measure is Measure


 * Co-Countable Measure is Probability Measure