Lightly mixing is closed under countable products (Q1107654)

Let (X,\(\mu)\) be a Lebesgue probability space. A measure-preserving map \(T:\quad X\to X\) is said to be lightly mixing iff \(\liminf_{n\to \infty}\mu (A\cap T^{-n}B)>0\) for any non-null sets \(A,B\subset X.\) The author shows that for any countable family \(\{T_ 1,T_ 2,...\}\) of lightly mixing transformations their direct product \(T_ 1\times T_ 2\times...\) is also lightly mixing. This is an answer to a question raised by N. A. Friedman. The following result plays a fundamental role in the proof of the main theorem: if each product \(T_ 1\times...\times T_ n\) is lightly mixing then the product \(T_ 1\times T_ 2\times...\) is lightly mixing. This reduction allows to confine ourselves to twofold products of lightly mixing maps.