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How To Make Davies' Theorem Visible
Author(s) -
Csörnyei Marianna
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/33.1.59
Subject(s) - mathematics , set (abstract data type) , plane (geometry) , element (criminal law) , point (geometry) , measure (data warehouse) , borel set , sigma , combinatorics , finite set , discrete mathematics , mathematical analysis , geometry , physics , computer science , quantum mechanics , data mining , political science , law , programming language
We prove that for an arbitrary measurable set A ⊂ R 2 and a σ‐finite Borel measure μ on the plane, there is a Borel set of lines L such that for each point in A , the set of directions of those lines from L containing the point is a residual set, and, moreover, μ ( Α ) = μ ( { ∪ l : l ∈ L } )We show how this result may be used to characterise the sets of the plane from which an invisible set is visible. We also characterise the rectifiable sets C 1 , C 2 for which there is a set which is visible from C 1 and invisible from C 2 .