Let E be a measurable set. $x$,$y$ $\in E$ are $\delta$ equivalent if $x=2^{n}y$ for some integer $n$. The $\delta$-index of a point $x$ in $E$ is the number of elements in its $\delta$ equivalent class and is denoted by $\delta{_{E}} (x)$. Let $E(\delta; k) = \{x\in E:\delta_{E}(x)=k\}$.Then E is the disjoint union of the sets $E(\delta; k)$.
My question is:
If E is a Lebesgue measurable set, then each $E(\delta,k)$, $(k\geq 1)$ is also Lebesgue measurable. How?
$E(\delta;k)=\cup_F \cap_{n \in F} (E\cap {2^{-n}} E)$ where the union is taken over all finite subsets of $\mathbb Z$ with cardinality $k$.