Let $m$ be Lebesgue measure. Find an example of Lebesgue measurable subsets $A_1, A_2, \dots$ of $[0,1]$ such that $m(A_n) > 0$ for each $n$, $m(A_n \Delta A_m) >0$ if $m \neq n$, and $m(A_n \cap A_m) = m(A_n)m(A_m)$ if $m\neq n$.
I've spent some time trying to come up with examples of sets that satisfy these conditions and here's what I've come up with so far:
These sets can't be properly nested since this would not satisfy the multiplicative property.
Each set must intersect every other set in order to satisfy the multiplicative property.
I've tried many variations of shifting shrinking closed intervals through the interval $[0,1]$.
Any nudges in the right direction would be appreciated.
Best Answer
Let $\sum_{k=1}^{\infty}\frac{\theta_k}{2^k}$ be the representation of the real number $\theta \in [0,1]$ in the binary system. For $n=1,2,\cdots$, we set $A_n=\{ \theta: \theta \in [0,1]\& \theta _n=0\}$. The $(A_n)_{n \in N}$ is a required sequence of subsets of $[0,1]$.