- Abstract Set Theory
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- Winter School in Abstract Analysis , section Set theory & Topology

However, it turns out that the concept of measurable function is not strong enough to solve some problems that arise later in function theory. For that reason, in , mathematicians found the family of uniform functions on a prescriptive space—a set with a fixed set of its finite covers. The importance and naturalness of the family of all uniform functions follows from the characterization theorem asserting that:. Since the times of H. Lebesgue and W. Young, there have been two parallel points of view in integration theory.

The first considers the integral as a special structure over a descriptive space with some measure. The second considers the integral as a superstructure over a functional linear space with some linear functional on it. Outstanding mathematicians spent many years trying to prove the parallelism of these two points of view. Solving both this problem and the general version took nearly one hundred years.

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## Abstract Set Theory

Riemann integrals have been generalized to arbitrary Tychonoff topological space with some bounded positive Radon measure. It is remarkable that the description of Riemann integrable functions requires families of uniform functions.

In , a characterization of Riemann integrable functions was discovered that is completely different from the famous Lebesgue characterization even for the real interval. Valeriy K.

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The extreme importance and naturalness of the family of all measurable functions follows from the famous Borel—Lebesgue—Hausdorff theorem asserting that: 1 this family is closed under all natural mathematical operations like addition, multiplication, and division and uniform convergence, and 2 every family of real-valued functions on a set with the mentioned properties is some family of all measurable functions on some descriptive space.

This family is extraordinarily abundant throughout the mathematical world.

The importance and naturalness of the family of all uniform functions follows from the characterization theorem asserting that: 1 this family is closed under all natural mathematical operations and uniform convergence, and 2 every family of real-valued bounded functions on a set with the mentioned properties is some family of all uniform functions on some prescriptive in particular, descriptive space. This family is also extraordinarily abundant. Timofey V.

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## Winter School in Abstract Analysis , section Set theory & Topology

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