Construction of a non-standard integral on AC [0, 1]




Ornas, Gerard

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A review of the Riemann and Lebesgue integration leads into the development and proof of the Riesz Representation Theorem. After this a new function space and a new norm are presented. A dense subset is extracted and used to approximate the functions in the new space. These approximations are used to define a nonstandard integral and an accompanying integral representation theorem. Chapter One is a review of the basic definitions needed throughout the remainder of the thesis. For the less common definitions an example function is provided. A very fundamental theorem, that the space C [0, 1] is complete is worked out in detail. The Riemann integral is defined, both as the supremum of Riemann sums over partitions, and as the limit of Riemann sums as the norm of the partition goes to zero. Finally, the Fundamental Theorem of Calculus is given. Chapter Two traces the development of the Lebesgue integral, following the path of H. L. Royden's Real Analysis. The chapter begins by showing the deficiencies of the Riemann integral. The first step in constructing the Lebesgue integral to avoid these difficulties is to define measure for sets. Next the Lebesgue integral is defined for a wider and wider range of functions. Starting with simple functions and moving up to the general case of Lebesgue integrable functions. The theorem that a function is the anti-derivative of another function if and only if it is absolutely continuous is stated. Finally, the chapter closes with an example of a very peculiar function, which illustrates the distinction between integration and antidifferentiation. Chapter Three also follows the path in Royden. More care is given however to show and fill in the proofs. The chapter is mostly a sequence of lemmas, definitions, and propositions, leading to the very important Riesz Representation Theorem. The chapter closes with the statement and a detailed proof, following that of Royden. Chapter Four is the actual development of the nonstandard, variation integral, developed by Edwards and Wayment. The chapter starts by setting up the space of absolutely continuous functions on [0, 1]. Next the norm is defined and proved to be a norm. Next the set of polygonal functions is shown to be dense in the space of absolutely continuous functions. Using this subset to approximate the absolutely continuous functions, the computable variation integral is defined. Simultaneously, an integral representation theorem for the new integral, representing the integrals as bounded linear functionals is developed. Finally, a couple of examples are given to illustrate the use of the integral.



Riemann integral, Lebesgue integral, integral theorems


Ornas, G. L. (1998). Construction of a non-standard integral on AC [0, 1](Unpublished thesis). Southwest Texas State University, San Marcos, Texas.


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