Co-Countable Measure is Probability Measure

Theorem

Let $X$ be an uncountable set.

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

Then the co-countable measure $\mu$ on $X$ is a probability measure.

Proof

By Co-Countable Measure is Measure, $\mu$ is a measure.

By Relative Complement with Self is Empty Set, have $\complement_X \left({X}\right) = \varnothing$.

As $\varnothing$ is countable, it follows that $X$ is co-countable.

Hence $\mu \left({X}\right) = 1$, and so $\mu$ is a probability measure.

$\blacksquare$