Quasi-invariant measures concentrating on countable structures

Countable L-structures N whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those N which have no algebraicity. Here we characterize those countable L-structure N whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those N which are not "highly algebraic" -- we say that N is highly algebraic if outside of every finite F there is some b and a tuple ā disjoint from b so that b has a finite orbit under the pointwise stabilizer of ā in Aut(N). As a bi-product of our proof we show that whenever the isomorphism class of N admits a quasi-invariant measure, then it admits one with continuous Radon--Nikodym cocycles.