ABSTRACT
Given a measure space https://www.w3.org/1998/Math/MathML"> ( S , E , μ ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math2_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and a countable collection of sets https://www.w3.org/1998/Math/MathML"> { A n } n = 1 ∞ ⊂ E , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math2_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> one is sometimes interested in the properties of sets related to https://www.w3.org/1998/Math/MathML"> ∩ n A n . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math2_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Of course https://www.w3.org/1998/Math/MathML"> ∩ A n ∈ E , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math2_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> but it can turn out that https://www.w3.org/1998/Math/MathML"> ∩ A n = ∅ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003260547/1539354b-4ff6-4752-881b-983ba2324ebc/content/math2_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> even when this collection has other significant non-empty intersection properties. For example, there may exist points x which are in infinitely many of these sets, and we could distinguish between the following:
Points x which are in infinitely many of these sets, and outside only finitely many.
Points x which are in infinitely many of these sets, and outside infinitely many.
