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 if and only if $X \setminus E$ is countable.

Also see

• Co-Countable Measure is Measure

• Co-Countable Measure is Probability Measure

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.7 \ \text{(ii)}$