Pathway in measure and integration theories
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The thesis is divided into three parts. Part One, Measure and Measure Spaces, contains the introduction; and in later chapters this part approaches measure theory by the most natural route. It begins in a detailed fashion by developing the theory of set length through several levels of generalization, including the concepts of ordinary interval length, Jordan content, and Lebesgue measure. Then, without details, it discusses the extension of this theory of measure to more complex Euclidean sets and, finally, to abstract sets. Part Two, The Integral on Measure Spaces, introduces the theory of Lebesgue integration. The integrals treated gere are the measure dependent Lebesgue and general Lebesgue integrals. Chapter 6 in this part establishes a link between the Riemann integral and Jordan content, described in Chapter 3 of Part One, and between the Lebesgue integral and Lebesgue measure, described in Chapter 4 of Part One. Chapter 7 establishes an analogous link between the general Lebesgue integral and general measure spaces introduced in Chapter 5 of Part One. Chapter 8 , Measure on a Semialgebra, stands as a bridge between the historic and abstract versions of the Lebesgue Theory; for it generalizes the procedure used to construct the geometric integral, and shows that this procedure will produce general integrals as well. Part Three, A More General Integral , considers the theory of the Daniell integral. This integral is defined in more general terms than the measure-dependent Lebesgue integral of Part Two. Nevertheless, a harmonious resolution is reached in the theorems that show how the lines supporting the two theories can be drawn together
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