Quasi-arithmetic-type invariant means on probability space

For a family $$(\mathscr {A}_x)_{x \in (0,1)}$$ of integral quasi-arithmetic means satisfying certain measurability-type assumptions we search for an mean K such that $$K\big ((\mathscr {A}_x(\mathbb {P}))_{x (0,1)}\big )=K(\mathbb {P})$$ every compactly supported probability Borel measure $$\mathbb {P}$$ . Also some results concerning the uniqueness invariant will be given.