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• # Artykuł - szczegóły

## Formalized Mathematics

2006 | 14 | 4 | 143-152

## Integral of Real-Valued Measurable Function1

EN

### Abstrakty

EN
Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

143-152

wydano
2006-01-01
online
2008-06-13

### Twórcy

autor
• Shinshu University, Nagano, Japan
autor
• Gifu National College of Technology, Gifu, Japan

### Bibliografia

• [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
• [3] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.
• [4] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.
• [5] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.
• [6] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21-26, 1996.
• [7] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• [8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• [9] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [10] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
• [11] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [12] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006.
• [13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.
• [14] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
• [15] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525-529, 2001.
• [16] P. R. Halmos. Measure Theory. Springer-Verlag, 1987.
• [17] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
• [18] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841-845, 1990.
• [19] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
• [20] 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.
• [21] Andrzej Nedzusiak. 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] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.
• [24] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
• [25] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.
• [26] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.