Measure Theory of Premeasures and Measures with Extension

Abstract

Measure theory is one of the important scenarios in mathematics as it has better properties as compared to set theory.  We aimed to focus on the premeasures and measures construction along with their extension. Initially the Lebesgue measure was used at undergraduate mathematical study but recently there are many measures are available such as Borel, Gauss Measure, Euler Measure, Hausdorff Measure, Haar Measure and many more.In construction of few measures the Carath´eodory extension theorem as well as Carath´eodory-Hahn theorem was solicited subsequently. Similarly the Dynkin’s π − λ theorem was also used in construction purpose. This paper illustrated the mathematical analysis for construction of measures and premeasures. Also extensions of measures and premeasures were studied in this paper.