An infinite sequence $α$ over an alphabet $Σ$ is $μ$-distributed w.r.t. a probability map $μ$ if, for every finite string $w$, the limiting frequency of $w$ in $α$ exists and equals $μ(w)$. %We raise the question of how to characterize the probability maps $μ$ for which $μ$-distributedness is preserved across finite-state selection, or equivalently, by selection by programs using constant space. We prove the following result for any finite or countably infinite alphabet $Σ$: every finite-state selector over $Σ$ selects a $μ$-distributed sequence from every $μ$-distributed sequence \emph{if and only if} $μ$ is induced by a Bernoulli distribution on $Σ$, that is a probability distribution on the alphabet extended to words by taking the product. The primary -- and remarkable -- consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which finite-state selection preserves $μ$-distributedness. The main positive takeaway is that (the appropriate generalization of) Agafonov's Theorem holds for Bernoulli distributions (rather than just equidistributions) on both finite and countably infinite alphabets. As a further consequence, we obtain a result in the area of symbolic dynamical systems: the shift-invariant measures $μ$ on $Σ^ω$ such that any finite-state selector preserves the property of genericity for $μ$, are exactly the positive Bernoulli measures.