Is there a subsequence of sets $\{A_{k_j}\}$ where the intersection is positive.

Question

Suppose $\{A_k\}_{k=1}^\infty$ a family of open subsets of $\mathbb{R}^n$ where

$$ A_k\subset B_1=\{x\in \mathbb{R}^n : |x|<1\}\,\,\,\mbox{and}\,\,\,\mu(A_k)\geq\epsilon>0. $$

Is there some subsequence $\{A_{k_j}\}_{j=1}^\infty\subset\{A_k\}_{k=1}^\infty$ such that

$$ \mu(\cap_{j=1}^\infty A_{k_j})> 0? $$

Answer 1

This is false. Consider $n=1$ and

$$A_1 = (0,1/2),$$

$$A_2 = (0,1/4)\cup (1/2,3/4),$$

$$A_3 = (0,1/8)\cup (1/4,3/8)\cup (1/2,5/8)\cup(3/4,7/8),$$

etc. Then, the intersection of any two sets $A_i$ and $A_j$ is necessarily $\min\{2^{-i-1},2^{-j-1}\}$.