Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

**Noboru Endou**- Shinshu University, Nagano
**Katsumi Wasaki**- Shinshu University, Nagano
**Yasunari Shidama**- Shinshu University, Nagano

- In this article we introduce some definitions concerning measurable functions and prove related properties.

- Cardinal Numbers of ${\Bbb Z}$ and ${\Bbb Q}$
- Basic Operations of Extended Real Valued Functions
- Level Sets
- Measurable Functions

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Grzegorz Bancerek.
Countable sets and Hessenberg's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [5]
Jozef Bialas.
Infimum and supremum of the set of real numbers. Measure theory.
*Journal of Formalized Mathematics*, 2, 1990. - [6]
Jozef Bialas.
Series of positive real numbers. Measure theory.
*Journal of Formalized Mathematics*, 2, 1990. - [7]
Jozef Bialas.
The $\sigma$-additive measure theory.
*Journal of Formalized Mathematics*, 2, 1990. - [8]
Jozef Bialas.
Several properties of the $\sigma$-additive measure.
*Journal of Formalized Mathematics*, 3, 1991. - [9]
Jozef Bialas.
Completeness of the $\sigma$-additive measure. Measure theory.
*Journal of Formalized Mathematics*, 4, 1992. - [10]
Jozef Bialas.
Some properties of the intervals.
*Journal of Formalized Mathematics*, 6, 1994. - [11]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama.
Basic properties of extended real numbers.
*Journal of Formalized Mathematics*, 12, 2000. - [16]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [17]
Andrzej Kondracki.
Basic properties of rational numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [19]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [20]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [21]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [22]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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