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Is there a translation-invariant way to assign probabilities to infinite subsets of $mathbb{Z}$?

Mathematics Asked on February 23, 2021

There is no uniform probability distribution over the integers. But I’m wondering whether the next best thing can be constructed. My question is, is there a meaningful way to assign probabilities to (some) infinite subsets of $mathbb{Z}$ which is translation-invariant?

I’m thinking asymptotic density, though I’m not sure if that’s translation-invariant or not.

One Answer

Asymptotic density is finitely additive and translation-invariant; see this PDF, for instance.

Correct answer by Brian M. Scott on February 23, 2021

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