A convergence law for continuous logic and continuous structures with finite domains

We consider continuous relational structures with finite domain [n] := {1, \ldots, n} and a many valued logic, CLA, with values in the unit interval and which uses continuous connectives and continuous aggregation functions. CLA subsumes first-order logic on ``conventional'' finite structures. To each relation symbol R and identity constraint ic on a tuple the length of which matches the arity of R we associate a continuous probability density function μ_R^{ic} : [0, 1] \to [0, \infty). We also consider a probability distribution on the set W_n of continuous structures with domain [n] which is such that for every relation symbol R, identity constraint ic, and tuple \bar{a} satisfying ic, the distribution of the value of R(\bar{a}) is given by μ_R^{ic}, independently of the values for other relation symbols or other tuples. In this setting we prove that every formula in CLA is asymptotically equivalent to a formula without any aggregation function. This is used to prove a convergence law for CLA which reads as follows for formulas without free variables: If φ\in CLA has no free variable and I \subseteq [0, 1] is an interval, then there is α\in [0, 1] such that, as n tends to infinity, the probability that the value of φ is in I tends to α.