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 defined by:


 * $\mu: \Sigma \to \overline{\R}, \ \mu \left({E}\right) := \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 iff $X \setminus E$ is countable.

Also see

 * Co-Countable Measure is Measure


 * Co-Countable Measure is Probability Measure