Example of Lebesgue measurable subsets satisfying conditions

Question

Let $m$ be Lebesgue measure. What is an example of Lebesgue measurable subsets $A_1, A_2, \ldots$ of $[0, 1]$ such that $m(A_n) > 0$ for each $n$, $m(A_n \Delta A_m) > 0$ if $n \neq m$, and $m(A_n \cap A_m) = m(A_n)m(A_m)$ if $n \neq m$?

Answer 1

You want sets that are independent in the sense of probability. Simple example: $$A_1=[0,1/2],$$ $$A_2=[0,1/4]\cup[1/2,3/4],$$ $$A_3=[0,1/8]\cup[1/4,3/8]\,\cup[1/2,5/8]\cup[3/4,7/8],$$ $$\dots$$