Consider a Borel action of a countable group G on a standard Borel space X. A countable Borel partition P of X is called a generator if GP = {gA : g ∈ G,A ∈ P} generates the Borel sigma-algebra of X. The existence of such P of cardinality n is equivalent to the existence of a G-embedding of X into the shift n G. For G = Z, the Kolmogorov Sinai

theorem implies that finite generators do not exist in the presence of an invariant probability measure with infinite entropy. It was asked by Weiss in the late 80s, whether the nonexistence of any invariant probability measure would guarantee the existence of a finite generator. We show that the answer is positive in case X admits a sigma-compact topological realization (e.g. if X is a sigma-compact Polish G-space). We also show that finite generators always exist in the context of Baire category thus answering a question of Kechris. More precisely, we prove that if X is an aperiodic Polish G-space, then there is a 4-generator on an invariant comeager set.