Group of unitary operators deducted from a spectral measure -- an application. (Q997976)

Let \(G\) and \(G'\) be locally compact abelian groups, assume that the duals \(G\) and \(G'\) have countable bases, let \(h: G\to G'\) be a continuous homomorphism, and let \((X_{g})_{g\in G}\) be a stationary continuous random function taking values in a Hilbert space and having a given associated random measure. The purpose of this note is to determine all stationary continuous random functions \((X_{g'}')_{g'\in G'}\) such that \((X_{h(g)}')_{g\in G}=(X_{g})_{g\in G}\).