A stability property for probability measures on Abelian groups (Q1579534)

A probability measure \(\mu\) on an Abelian locally compact group \(G\) is said to have a trivial equivalence class if it is uniquely defined, up to a shift and a central symmetry, by the modulus of its characteristic function. Let \(\mu_1\) be a probability measure on \(\mathbb R\) whose characteristic function is an entire function of finite order with real zeroes. It is shown that if \(\mu\) has a trivial equivalence class, then the same property holds for \(\mu_1\times \mu\) on \(\mathbb R\times G\). Conditions are found under which similar results are valid with a Gaussian measure on a general group or some measures on \(\mathbb Z\) used instead of a measure on \(\mathbb R\).