Co-Countable Measure is Probability Measure
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Theorem
Let $X$ be an uncountable set.
Let $\AA$ 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 $\relcomp X X = \O$.
As $\O$ is countable, it follows that $X$ is co-countable.
Hence $\map \mu X = 1$, and so $\mu$ is a probability measure.
$\blacksquare$