L-subalgebras of measures related to dissociate sets
Let E be a dissociate (or independent) Borel subset of the LCA group G. Let \(G_ p'E=\{\sum^{n}_{1}\pm x_ j:\quad n\geq 1,\quad x_ j\in E,\quad x_ j\neq x_ k\}.\) The (somewhat technical) main result of the paper implies the following. There exists a family of probability measures \(F\subseteq M_ c(G_ p'E)\) such that (i) if \(y\in G\), \(k\geq 1 (m_ 1,...,m_ k)\) and \((p_ 1,...,p_ k)\) are k-tuples of non- negative integers, and \(\mu_ 1,...,\mu_ k\in F,\) then \(\delta_ y*\mu_ 1^{m_ 1}*...*\mu_ k^{m_ k}\) is mutually singular with respect to \(\mu_ 1^{p_ 1}*...*\mu_ k^{p_ k}\) unless \(y=0\) and \((m_ 1,...,m_ k)=(p_ 1,...,p_ k)\) and (ii) \(M_ c(G_ p'E)\) is generated, as an L-subalgebra, by F. Results of \textit{E. Hewitt} and \textit{S. Kakutani} [ibid. 4, 553-574 (1960; Zbl 0100.118)] and others are thereby extended. The novelty of the present paper lies in a permutation argument and an explicit use of the sum mapping from \(G\times...\times G\to G\).
scientific article; zbMATH DE number 3863821