A homomorphism of minimal
flows XY , has a relatively invariant measure if there exists a positive projection
from 𝒞(X) onto 𝒞(Y ) which commutes with translasion. Such a relatively invariant
measure does not always exists. However, some elementary facts from the theory of
compact convex sub-sets of a locally convex topological vector space are used to show
that given a homomorphism of minimal flows XY there exists a commutative
diagram
where 𝜃 and 𝜃∼ are strongly proximal homomorphisms and ϕ∼ has a relatively invariant
measure, (RIM). Homomorphisms which have invariant measures are studied and
questions of existence and uniqueness are investigated.
