Integral of Complex-Valued Measurable Function

Keiko Narita; Noboru Endou; Yasunari Shidama

Formalized Mathematics (2008)

  • Volume: 16, Issue: 4, page 319-324
  • ISSN: 1426-2630

Abstract

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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

How to cite

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Keiko Narita, Noboru Endou, and Yasunari Shidama. "Integral of Complex-Valued Measurable Function." Formalized Mathematics 16.4 (2008): 319-324. <http://eudml.org/doc/266613>.

@article{KeikoNarita2008,
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},
author = {Keiko Narita, Noboru Endou, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {formalization of Riemann integral},
language = {eng},
number = {4},
pages = {319-324},
title = {Integral of Complex-Valued Measurable Function},
url = {http://eudml.org/doc/266613},
volume = {16},
year = {2008},
}

TY - JOUR
AU - Keiko Narita
AU - Noboru Endou
AU - Yasunari Shidama
TI - Integral of Complex-Valued Measurable Function
JO - Formalized Mathematics
PY - 2008
VL - 16
IS - 4
SP - 319
EP - 324
AB - 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
LA - eng
KW - formalization of Riemann integral
UR - http://eudml.org/doc/266613
ER -

References

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  1. [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
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  15. [15] Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006. Zbl1298.26030
  16. [16] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990. 
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Citations in EuDML Documents

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  1. Yasushige Watase, Noboru Endou, Yasunari Shidama, On L 1 Space Formed by Complex-Valued Partial Functions
  2. Keiko Narita, Noboru Endou, Yasunari Shidama, Lebesgue's Convergence Theorem of Complex-Valued Function
  3. Hiroyuki Okazaki, Yasunari Shidama, Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables
  4. Keiko Narita, Noboru Endou, Yasunari Shidama, The Measurability of Complex-Valued Functional Sequences
  5. Hiroyuki Okazaki, Yasunari Shidama, Probability on Finite Set and Real-Valued Random Variables
  6. Yasushige Watase, Noboru Endou, Yasunari Shidama, On L p Space Formed by Real-Valued Partial Functions

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