Why isn't natural density $D$ countably additive on the algebra $\mathcal{A}$ generated by the periodic sets $\{n, 2n, 3n, \ldots\}$ (where $n$ is a natural number)? I tried finding examples like $D(\{1\}) + D(\{2\}) + D(\{3\}) + \cdots \ne D(\{1, 2, 3, \ldots\})$ in $\mathcal{A}$, but they don't exist: for every element $S$ of $\mathcal{A}$, there exists a natural number $n$ such that $x \in S$ if and only if $x + n \in S$. Therefore, $S$ is empty or has density greater than $0$.

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    $\begingroup$ Since any two of those periodic sets will intersect, I'm not sure how countable additivity can be formulated at all (that property involves disjoint sets). $\endgroup$ Sep 20 at 6:33


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