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## Measurable sets

Let $\{A_n\}_{n \in \mathbb{N}}$ be a sequence of measurable subsets of the real line which covers almost every every point infinitely often. Prove that there exists a set $B\subset \mathbb{N}$ of zero density ($\displaystyle \lim_{n\to \infty} \frac{|B\cap \{0,1,...,n-1\}|}{n}=0$) such that $\{A_n\}_{n \in B}$ also covers every point infinitely often.
Miklos Schweitzer 2009 Problem 8

I think I might have found a generalization of the problem. My approach is in the pdf attached.
Problem 8 – Miklos Schweitzer 2009