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I'm having difficulty finding an example of a strictly positive probability measure on $\mathbb{Z}$. The closest I've managed to get is information about finitely additive ℤ-invariant probability measures on ℤ and a very vague question which asks for information about a proof that a probability measure on ℤ with some given properties.

Are there any "nice" examples of such a measure? If not, can anyone point me to a reference with any examples? Thank you!

MathFrak96
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    Take any convergent series of positive terms, like $1+\frac{1}{2}+\frac{1}{4}+\cdots=2$. Divide by the sum's value to normalize, like $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots=1$. Then split all the terms after the first one in half and assign them to negative indices as well, like $\cdots+\frac{1}{16}+\frac{1}{8}+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\cdots=1$. – coiso Jun 02 '23 at 18:43
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    Choose your favorite convergent series $s=\sum_{n\in\mathbb{Z}}a_n$ with $a_n>0$. Then $\frac{1}{s}\sum_na_n\delta_n$ is a probability law on $\mathbb{Z}$ with the desired property. – Mittens Jun 02 '23 at 19:05

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Start with $\mu(n)=2^{-|n|}$.

Well, it's not a probability measure because $\sum_{n \in \mathbb Z} 2^{-|n|} = 3$.

So now change it to $\mu(n) = \frac{1}{3} \cdot 2^{-|n|}$.

FShrike
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Lee Mosher
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