Multiple disjointness and invariant measures on minimal distal flows

As the main theorem, it is proved that a collection of minimal $PI$-flows with a common phase group and satisfying a certain algebraic condition is multiply disjoint if and only if the collection of the associated maximal equicontinuous factors is multiply disjoint. In particular, this result holds for collections of minimal distal flows. The disjointness techniques are combined with Furstenberg's example of a minimal distal system with multiple invariant measures to find the exact cardinalities of (extreme) invariant means on $\mathcal{D}(\mathbb{Z})$ and $\mathcal{D}(\mathbb{R})$, the spaces of distal functions on $\mathbb{Z}$ and $\mathbb{R}$, respectively. In all cases, this cardinality is $2^{\mathfrak{c}}$. The size of the quotient of $\mathcal{D}(\mathbb{Z})$ or of $\mathcal{D}(\mathbb{R})$ by a closed subspace with a unique invariant mean is observed to be non-separable by applying the same ideas.