A Review of the Construction of Particular Measures

Abstract

The first measure one usually comes into contact with in undergraduate mathematical studies is the Lebesgue measure and seeing how it is applied to the Lebesgue integral to extend considerably the Riemann integral, it doesn’t take very much else to arouse one’s interest in the study of measures and their construction with the hope/intent of eliciting their usefulness and how they are applied to other areas of mathematics. The Carath´eodory extension theorem and the Carath´eodory-Hahn theorem which are invoked subsequently in the construction of some measures are stated without proof. A large class of measures exist and this paper illustrates the construction of some of these measures including the Radon measure, the Hausdorff measure, the Lebesgue-Stieltjes measure, the Lebesgue measure in Rn and Product measures. The material presented is standard but it provides a summary of some key points on measure theory which might prove to be useful for the undergraduate.