It is well known that the asymptotic density of an infinite union of disjoint sets of integers may not be the sum of their individual asymptotic densities.


Can this failure of countable additivity occur even if each of the sets being unioned is periodic (calling a set periodic if there is some nonzero integer such that it is closed under addition and subtraction of this integer)?



Additivity of asymptotic density of periodic sets
For each $r\in\mathbb{Z}$, let $a\_{r}$ be a positive real number so that $\sum\_{r\in\mathbb{Z}}a\_{r}<1$. Let $A\_{r}\subseteq\mathbb{Z}$ be a periodic set with $\mu(A\_{r})<a\_{r}$ and $r\in A\_{r}$. Then $\mathbb{Z}=\bigcup\_{r\in\mathbb{Z}}A\_{r}$. Now reenumerate the set $\{A\_{r}|r\in\mathbb{R}\}=\{B\_{n}|n\in\mathbb{N}\}$. Then $\sum\_{n}\mu(B\_{n})<1$. Let $C\_{n}=B\_{n}\setminus(B\_{0}\cup...\cup B\_{n-1})$. Then $\mathbb{Z}=\bigcup\_{n}C\_{n}$, but $\sum\_{n}\mu(C\_{n})\leq\sum\_{n}\mu(B\_{n})<1$, so this measure is not countably additive.


