We study the relationship between logical and behavioral equivalence
for coalgebras on general measurable spaces. Modal logics are
interpreted using predicate liftings. Prominent examples of these
coalgebras include stochastic relations and labelled Markov
transition systems. The corresponding Hennessy--Milner type logics are a prime example for coalgebraic modal logics.  

It is shown that the notions of logical und behavioral equivalence
coincide for coalgebras for a wide class of functors. Moreover, we
present some results on compositionality of the logics. Throughout, we
establish our results for general measurable spaces without relying on
topological assumptions.