  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.
