
Accession Number : AD0711294
Title : OUTER MEASURE, BOREL SETS AND LEBESGUE MEASURE IN THE PLANE.
Descriptive Note : Master's thesis,
Corporate Author : NAVAL POSTGRADUATE SCHOOL MONTEREY CALIF
Personal Author(s) : Heming,David Millar
Report Date : JUN 1970
Pagination or Media Count : 46
Abstract : The essential properties of general Lebesgue outer measure are discussed. The complete measure space, consisting of the general Lebesgue outer measure restricted to the measurable sets, is developed and this measure is shown to be unique. Two characterizations of measurable sets are discussed. The Borel sets are investigated in the plane and more generally, in nspace, and it is shown that the sigmaalgebra of Borel sets is equal to the product sigmaalgebra of Borel sets on the line. Finally, the interrelationships between Lebesgue measure in the plane and the product measure of Lebesgue measures on the line are investigated. It is shown that the sigmaalgebra of Lebesgue measurable sets properly contains the product sigmaalgebra and that these two measures agree on the product sigmaalgebra. It is also proven that the sigmaalgebra of Lebesgue measurable sets is the completion of the product sigmaalgebra. Examples are provided to illustrate that the product measure spaces discussed are not complete as well as an example of a subset of the plane which is not Lebesgue measurable. (Author)
Descriptors : (*MEASURE THEORY, THEOREMS), SET THEORY, NUMERICAL INTEGRATION, THESES
Subject Categories : Theoretical Mathematics
Distribution Statement : APPROVED FOR PUBLIC RELEASE