Definition:Co-Countable Measure

Theorem
Let $X$ be an uncountable set, and 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, and $E$ is co-countable iff $X \setminus E$ is countable.

Also see
That $\mu$ actually is a measure is shown on Co-Countable Measure is Measure.

In fact, a co-countable measure is a probability measure.