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An infinite sequence $\alpha$ over an alphabet $\Sigma$ is $\mu$-distributedw.r.t. a probability map $\mu$ if, for every finite string $w$, the limitingfrequency of $w$ in $\alpha$ exists and equals $\mu(w)$. %We raise the questionof how to characterize the probability maps $\mu$ for which$\mu$-distributedness is preserved across finite-state selection, orequivalently, by selection by programs using constant space. We prove thefollowing result for any finite or countably infinite alphabet $\Sigma$: everyfinite-state selector over $\Sigma$ selects a $\mu$-distributed sequence fromevery $\mu$-distributed sequence \emph{if and only if} $\mu$ is induced by aBernoulli distribution on $\Sigma$, that is a probability distribution on thealphabet extended to words by taking the product. The primary -- and remarkable-- consequence of our main result is a complete characterization of the set ofprobability maps, on finite and infinite alphabets, for which finite-stateselection preserves $\mu$-distributedness. The main positive takeaway is that(the appropriate generalization of) Agafonov's Theorem holds for Bernoullidistributions (rather than just equidistributions) on both finite and countablyinfinite alphabets. As a further consequence, we obtain a result in the area ofsymbolic dynamical systems: the shift-invariant measures $\mu$ on$\Sigma^{\omega}$ such that any finite-state selector preserves the property ofgenericity for $\mu$, are exactly the positive Bernoulli measures.

Agafonov's Theorem for finite and infinite alphabets and probability distributions different from equidistribution