An elementary remark on the intersection of sets.

Abstract

In this paper, we will investigate the following question: Given $ C \in (0,1) $ and a sequence $ A_n \subseteq [0,1] $ with $ \lambda(A_n)=C $, when does there exist a subsequence $ A_{n_i} $ such that $ \lambda( \cap_i A_{n_i} ) >0 $? We will show that the answer to this question can be characterized by the properties of a function $g$ which will be a weak $ L^1 $ limit of characteristic functions.