A Review of the Construction of Particular Measures
William Obeng-Denteh
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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 Carathéodory extension theorem and the Carathéodory-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 R n 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.
Key takeaways
- Then the collection M of sets that are measurable with respect to µ * is a σ-algebra.
- Then the Carathéodory measure µ induced by µ is an extension of µ.
- By the product measure m = µ × v, we mean the Carathéodory extension of the premeasure m : R → [0, ∞] defined on the σ-algebra of (µ × v) * -measurable subsets of X × Y as stipulated in the Carathéodory-Hahn theorem.
- A Borel measure µ is called a Radon measure provided, To arrive at the definition above, a natural place to begin is to consider premeasures µ : T → [0, ∞] defined on the topology T and consider the Carathéodory outer measure induced by µ.
- The restriction of µ * n to L n is called the Lebesgue measure on R n or n-dimensional Lebesgue measure and denoted by µn.
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