On a problem of I. Z. Ruzsa (Q2563793)

Let \(G\) be a topological Abelian group, \(\mu_n\) be a sequence of probability measures on \(G\). A sequence \(\nu_n\) of probability measures on \(G\) is called associated to \(\mu_n\) if \(\mu_n= \nu_n* \delta_{a_n}\), where \(a_n\in G\) and \(\delta_x\) is the degenerate probability measure concentrated at the point \(x\in G\). A product of probability measures is called absolutely convergent if each of its rearrangements converges to the same limit. I. Z. Ruzsa proved that if \(G\) is a compact Abelian group, then every convergent product is associated to an absolutely convergent one. Using this result the author proves that Ruzsa's theorem is valid on arbitrary LCA groups.