Quasi-arithmetic-type invariant means on probability space.

For a family (A x) x ∈ (0 , 1) of integral quasi-arithmetic means satisfying certain measurability-type assumptions we search for an integral mean K such that K ( (A x (P)) x ∈ (0 , 1) ) = K (P) for every compactly supported probability Borel measure P . Also some results concerning the uniqueness of invariant means will be given. [ABSTRACT FROM AUTHOR]