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Zero density subseq. of set sequence covers R a.e. i.often

Source: Schweitzer 2009

November 13, 2009
limitprobabilityreal analysiscollege contestsMiklos Schweitzer

Problem Statement

Let {An}n∈N \{A_n\}_{n \in \mathbb{N}} be a sequence of measurable subsets of the real line which covers almost every point infinitely often. Prove, that there exists a set B⊂N B \subset \mathbb{N} of zero density, such that {An}n∈B \{A_n\}_{n \in B} also covers almost every point infinitely often. (The set B⊂N B \subset \mathbb{N} is of zero density if \lim_{n \to \infty} \frac {\#\{B \cap \{0, \dots, n \minus{} 1\}\}}{n} \equal{} 0.)
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