A theorem due to Hindman states that if E is a subset of ? with d*(E) > 0, where d* denotes the upper Banach density, then for any ? 0 there exists N ? such $$d^{\ast}(\cup_{i=1}^{N}(E-i))>1-\varepsilon$$ . Curiously, this result does not hold one replaces density $$\bar{d}$$ Originally proved combinatorially, Hindman’s allows quick and easy proof using an ergodic version Furstenberg’s correspo...

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.