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