Invariant measures for skew products and uniformly distributed sequences

'Almost all sequences (r ₁, ..., r _n, ...) of positive integers have the following "universal" property: Whenever (X, μ) is a Borel probability compact metric space, and Φ ₁, Φ ₂, ..., Φ _n, ... a sequence of commuting measure preserving continuous maps on (X, μ), such that the action (by composition) on (X, μ) of the semigroup with generators Φ ₁, ..., Φ _n, ... is uniquely ergodic and equicontinuous, then for every x ∈ X the sequence w ₁,w ₂, ..., w _n, ... where, is uniformly distributed for μ. This is a contribution to Problem 116 of Schreier and Ulam in the Scottish Book.