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

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.