Vol. 58, No. 2, 1975

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Relatively invariant measures

Shmuel Glasner

Vol. 58 (1975), No. 2, 393–410

A homomorphism of minimal flows X→ϕ Y , 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 Xϕ
→ Y there exists a commutative diagram

  ∼  𝜃∼
X∼   →    X
ϕ ↓∼      ↓ ϕ
Y    →𝜃   Y

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.

Mathematical Subject Classification 2000
Primary: 54H20
Secondary: 28A65
Received: 5 February 1974
Published: 1 June 1975
Shmuel Glasner