Elementary invariants for quantified probability logic

Let QPL be the two-sorted probabilistic language proposed in [8], which expands the well-known “polynomial” language described in [3, Section 6] by adding quantifiers over events. We show that all atomless spaces have the same QPL-theory, and this theory is decidable. Also we introduce the notion of elementary invariant for QPL and use it for obtaining exact complexity upper bounds for some interesting probabilistic theories.