# 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

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topKeiko 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

top- [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [2] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.
- [3] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.
- [4] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21-26, 1996.
- [5] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
- [6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [8] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006.
- [9] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.
- [10] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
- [11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
- [12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
- [13] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.
- [14] Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.
- [15] Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006. Zbl1298.26030
- [16] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
- [17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [18] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [19] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

## Citations in EuDML Documents

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

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