Convergence and 0-1 Laws for L_inf, omega^k under Arbitrary Measures

We prove some general results about the existence of 0–1 and convergence laws for L ∞,ω k and L ∞,ω k on classes of finite structures equipped with a sequence of arbitrary probability measures {μ n }, as well as a few results for particular classes. First, two new proofs of the characterization theorem of Kolaitis and Vardi [9] are given. Then this theorem is generalized to obtain a characterization of the existence of L ∞,ω ω convergence laws on a class with arbitrary measure. We use this theorem to obtain some results about the nonexistence of L ∞,ω ω convergence laws for particular classes of structures. We also disprove a conjecture of Tyszkiewicz [16] relating the existence of L ∞,ω ω and MSO 0–1 laws on classes of structures with arbitrary measures.