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2009 | 17 | 2 | 137-145
Tytuł artykułu

Lebesgue's Convergence Theorem of Complex-Valued Function

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their properties. In addition, we proved properties of complex-valued simple functions.
Słowa kluczowe
Wydawca
Rocznik
Tom
17
Numer
2
Strony
137-145
Opis fizyczny
Daty
wydano
2009-01-01
online
2009-07-14
Twórcy
autor
  • Hirosaki-city, Aomori, Japan
autor
  • Gifu National College of Technology, Japan
  • Shinshu University, Nagano, Japan
Bibliografia
  • [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] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
  • [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
  • [6] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  • [7] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  • [8] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008, doi:10.2478/v10037-008-0023-1.[Crossref]
  • [9] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006, doi:10.2478/v10037-006-0008-x.[Crossref]
  • [10] 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]
  • [11] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
  • [12] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
  • [13] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
  • [14] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
  • [15] 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]
  • [16] Keiko Narita, Noboru Endou, and Yasunari Shidama. The measurability of complex-valued functional sequences. Formalized Mathematics, 17(2):89-97, 2009, doi: 10.2478/v10037-009-0010-1.[Crossref]
  • [17] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
  • [18] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.
  • [19] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
  • [20] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.
  • [21] Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.
  • [22] 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]
  • [23] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403-410, 1997.
  • [24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  • [25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
  • [26] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_v10037-009-0015-9
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