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For a continuous action of a countable discrete group $G$ on a Polish space$X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P:= \{ gC : g \in G, C \in P \}$ generates the Borel $\sigma$-algebra of $X$.For $G = Z$, the Kolmogorov--Sinai theorem gives a measure-theoreticobstruction to the existence of finite generators: they do not exist in thepresence of an invariant probability measure with infinite entropy. It wasasked by Benjamin Weiss in the late 80s whether the nonexistence of anyinvariant probability measure guarantees the existence of a finite generator.We show that the answer is positive (in fact, there is a 32-generator) for anarbitrary countable group $G$ and $\sigma$-compact $X$ (in particular, forlocally compact $X$). We also show that any continuous aperiodic action of $G$on an arbitrary Polish space admits a 4-generator on a comeager set, thusgiving a positive answer to a question of Alexander Kechris asked in themid-90s.  Furthermore, assuming a positive answer to Weiss's question for arbitraryPolish spaces and $G = Z$, we prove the following dichotomy: every aperiodicBorel action of $Z$ on a Polish space $X$ admits either an invariantprobability measure of infinite entropy or a finite generator. As an auxiliarylemma, we prove the following statement, which may be of independent interest:every aperiodic Borel action of a countable group $G$ on a Polish space $X$admits a $G$-equivariant Borel map to the aperiodic part of the shift action of$G$ on $2^G$.  We also obtain a number of other related results, among which is a criterionfor the nonexistence of non-meager weakly wandering sets for continuous actionsof $Z$. A consequence of this is a negative answer to a question asked byEigen--Hajian--Nadkarni, which was also independently answered by BenjaminMiller.

Finite generators for countable group actions in the Borel and Baire category settings