Outer measure, Borel sets and Lebesgue measure in the plane
Heming, David Millar
Wilde, Carroll O.
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In this paper, 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 inves- tigated in the plane and more generally, in n-space, and it is shown that the a-algebra of Borel sets is equal to the product a-algebra 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 a-algebra of Lebesgue measurable sets properly contains the product a-algebra and that these two measures agree on the product a-algebra. It is also proven that the a-algebra of Lebesgue measurable sets is the completion of the product a-algebra. 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.
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