# The Measurability of Complex-Valued Functional Sequences

Keiko Narita; Noboru Endou; Yasunari Shidama

Formalized Mathematics (2009)

- Volume: 17, Issue: 2, page 89-97
- ISSN: 1426-2630

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topKeiko Narita, Noboru Endou, and Yasunari Shidama. "The Measurability of Complex-Valued Functional Sequences." Formalized Mathematics 17.2 (2009): 89-97. <http://eudml.org/doc/266849>.

@article{KeikoNarita2009,

abstract = {In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions. In addition, we proved properties of complex-valued simple functions.},

author = {Keiko Narita, Noboru Endou, Yasunari Shidama},

journal = {Formalized Mathematics},

keywords = {formalization of Riemann integral},

language = {eng},

number = {2},

pages = {89-97},

title = {The Measurability of Complex-Valued Functional Sequences},

url = {http://eudml.org/doc/266849},

volume = {17},

year = {2009},

}

TY - JOUR

AU - Keiko Narita

AU - Noboru Endou

AU - Yasunari Shidama

TI - The Measurability of Complex-Valued Functional Sequences

JO - Formalized Mathematics

PY - 2009

VL - 17

IS - 2

SP - 89

EP - 97

AB - In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions. In addition, we proved properties of complex-valued simple functions.

LA - eng

KW - formalization of Riemann integral

UR - http://eudml.org/doc/266849

ER -

## References

top- [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
- [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [4] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.
- [5] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.
- [6] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.
- [7] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21-26, 1996.
- [8] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
- [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [13] Wenpai Chang, Hiroshi Yamazaki, and Yatsuka Nakamura. The inner product and conjugate of finite sequences of complex numbers. Formalized Mathematics, 13(3):367-373, 2005.
- [14] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006, doi:10.2478/v10037-006-0008-x.[Crossref]
- [15] Noboru Endou, Yasunari Shidama, and Keiko Narita. Egoroff's theorem. Formalized Mathematics, 16(1):57-63, 2008, doi:10.2478/v10037-008-0009-z.[Crossref] Zbl1298.46005
- [16] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
- [17] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
- [18] Jarosław Kotowicz and Yuji Sakai. Properties of partial functions from a domain to the set of real numbers. Formalized Mathematics, 3(2):279-288, 1992.
- [19] Keiko Narita, Noboru Endou, and Yasunari Shidama. Integral of complex-valued measurable function. Formalized Mathematics, 16(4):319-324, 2008, doi:10.2478/v10037-008-0039-6.[Crossref] Zbl1298.26030
- [20] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
- [21] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.
- [22] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
- [23] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.
- [24] Yasunari Shidama and Noboru Endou. Integral of real-valued measurable function. Formalized Mathematics, 14(4):143-152, 2006, doi:10.2478/v10037-006-0018-8.[Crossref] Zbl1298.26030
- [25] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403-410, 1997.
- [26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [27] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [28] Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007, doi:10.2478/v10037-007-0026-3.[Crossref]

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