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General integration and extensions. I

Štefan Schwabik (2010)

Czechoslovak Mathematical Journal

A general concept of integral is presented in the form given by S. Saks in his famous book Theory of the Integral. A special subclass of integrals is introduced in such a way that the classical integrals (Newton, Riemann, Lebesgue, Perron, Kurzweil-Henstock...) belong to it. A general approach to extensions is presented. The Cauchy and Harnack extensions are introduced for general integrals. The general results give, as a specimen, the Kurzweil-Henstock integration in the form of the extension of...

General integration and extensions.II

Štefan Schwabik (2010)

Czechoslovak Mathematical Journal

This work is a continuation of the paper (Š. Schwabik: General integration and extensions I, Czechoslovak Math. J. 60 (2010), 961–981). Two new general extensions are introduced and studied in the class 𝔗 of general integrals. The new extensions lead to approximate description of the Kurzweil-Henstock integral based on the Lebesgue integral close to the results of S. Nakanishi presented in the paper (S. Nakanishi: A new definition of the Denjoy’s special integral by the method of successive approximation,...

Integral of Complex-Valued Measurable Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valued measurable functions is considered.MML identifier: MESFUN6C, version: 7.9.01 4.101.1015

Integral of Real-Valued Measurable Function 1

Yasunari Shidama, Noboru Endou (2006)

Formalized Mathematics

Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

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