Topologically equivalent measures in the Cantor space. II (Q1089450)

A C-pair is a pair (X,\(\mu)\) in which X is the product space \(\prod_{n\geq 1}S_ n\) and \(\mu\) is the product measure \(\prod_{n\geq 1}\mu_ n,\) where each \(S_ n\) is a finite set and \(\mu_ n\) is a probability measure on \(S_ n\). Let \(t\geq 2\) be a positive integer and \(p_ 1,p_ 2,...,p_ t\) nonnegative numbers such that \(p_ 1+p_ 2+...+p_ t=1.\) A C-pair \((X,\mu)\) is said to be of type \((t;p_ 1,p_ 2,...,p_ t)\) if each \(S_ n\) consists of exactly t elements which, for convenience, may be taken to be \(\{1,2,...,t\},\) and \(\mu_ n(\{j\})=p_ j,j=1,2,...,t.\) The following is one of the results proved in this paper. Theorem: Let \((X,\mu)\) be a C-pair \((s;q_ 1,q_ 2,...,q_ s)\) and \((Y,\nu)\) a C-pair \((t;p_ 1,p_ 2,...,p_ t).\) In order for \((X,\mu)\) to be isomorphic to (Y,\(\nu)\), i.e., if there is a homeomorphism h from X onto Y such that \(\mu (B)=\nu (h(B))\) for every Borel subset B of X, it is sufficient that there exist disjoint special clopen sets \(U_ 1,U_ 2,...,U_ t\) in X such that X \(= the\) union of these t sets and \(\mu (U_ j)=p_ j\) for all j. The assumption that the sets \(U_ j\) be special clopen sets (pencils) cannot be dropped in the above theorem. As an example, the C-pairs \((2;1/3,2/3)\) and \((2;1/4,3/4)\) are not isomorphic. As an illustration of the above theorem, it turns out that the C-pairs \((3;1/3,2/9,4/9)\) and \((3;1/9,2/9,2/3)\) are isomorphic. The necessity of the condition in the above theorem is also examined. [For Part I see Proc. Am. Math. Soc. 77, 229-236 (1979; Zbl 0386.28009).]