Integral of Measurable Function 1

 Access to full text

 Full (PDF)

1. [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
2. [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
3. [3] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991. 
4. [4] Józef Białcs. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991. 
5. [5] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991. 
6. [6] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21-26, 1996. 
7. [7] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245-254, 1990. 
8. [8] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990. 
9. [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990. 
10. [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
11. [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
12. [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990. 
13. [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990. 
14. [14] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001. 
15. [15] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001. 
16. [16] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525-529, 2001. 
17. [17] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Some properties of extended real numbers operations: abs, min and max. Formalized Mathematics, 9(3):511-516, 2001. 
18. [18] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990. 
19. [19] Grigory E. Ivanov. Definition of convex function and Jensen's inequality. Formalized Mathematics, 11(4):349-354, 2003. 
20. [20] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841-845, 1990. 
21. [21] 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. 
22. [22] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990. 
23. [23] Andrzej Nedzusiak. Probability. Formalized Mathematics, 1(4):745-749, 1990. 
24. [24] Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990. 
25. [25] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992. 
26. [26] Yasunari Shidama and Noboru Endou. Lebesgue integral of simple valued function. Formalized Mathematics, 13(1):67-71, 2005. 
27. [27] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics. 
28. [28] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990. 
29. [29] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990. 
30. [30] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003. 
31. [31] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990. 
32. [32] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
33. [33] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. 
34. [34] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. 

1. Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, Hiroyuki Okazaki, Inferior Limit, Superior Limit and Convergence of Sequences of Extended Real Numbers
2. Yasunari Shidama, Noboru Endou, Integral of Real-Valued Measurable Function 1
3. Keiko Narita, Noboru Endou, Yasunari Shidama, The First Mean Value Theorem for Integrals
4. Yasushige Watase, Noboru Endou, Yasunari Shidama, On L 1 Space Formed by Real-Valued Partial Functions
5. Keiko Narita, Noboru Endou, Yasunari Shidama, Integral of Complex-Valued Measurable Function
6. Noboru Endou, Keiko Narita, Yasunari Shidama, Fatou's Lemma and the Lebesgue's Convergence Theorem
7. Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama, Hopf Extension Theorem of Measure
8. Noboru Endou, Yasunari Shidama, Keiko Narita, Egoroff's Theorem
9. Noboru Endou, Keiko Narita, Yasunari Shidama, The Lebesgue Monotone Convergence Theorem
10. Keiko Narita, Noboru Endou, Yasunari Shidama, Lebesgue's Convergence Theorem of Complex-Valued Function