Integral of Complex-Valued Measurable Function

Keiko Narita; Noboru Endou; Yasunari Shidama

doi:10.2478/v10037-008-0039-6

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Integral of Complex-Valued Measurable Function

Formalized Mathematics

Volume 16 (2008): Issue 4 (December 2008)

By: Keiko Narita, Noboru Endou and Yasunari Shidama

Open Access

|Mar 2009

Abstract

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

References

[1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
[2] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.
Search in Google Scholar
[3] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.
Search in Google Scholar
[4] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21-26, 1996.
Search in Google Scholar
[5] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
Search in Google Scholar
[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
[7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[8] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006.10.2478/v10037-006-0008-x
Search in Google Scholar
[9] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.
Search in Google Scholar
[10] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
Search in Google Scholar
[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
Search in Google Scholar
[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
Search in Google Scholar
[13] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.
Search in Google Scholar
[14] Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.
Search in Google Scholar
[15] Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006.10.2478/v10037-006-0018-8
Search in Google Scholar
[16] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
Search in Google Scholar
[17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[18] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
Search in Google Scholar
[19] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Search in Google Scholar