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 prove the measurablility of some extended real valued functions which are $f$+$g$, $f$\,-\,$g$ and so on. Moreover, we will define the simple function which are defined on the sigma field. It will play an important role for the Lebesgue integral theory.

- Finite Valued Function
- Measurability of $f+g$ and $f - g$
- Definitions of Extended Real Valued Functions max$_{+}$($f$) and max$_{-}$($f$) and their Basic Properties
- Measurability of max$_{+}$($f$), max$_{-}$($f$) and $|f|$
- Definition and Measurability of Characteristic Function
- Definition and Measurability of Simple Function

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Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
Zermelo theorem and axiom of choice.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Jozef Bialas.
Infimum and supremum of the set of real numbers. Measure theory.
*Journal of Formalized Mathematics*, 2, 1990. - [5]
Jozef Bialas.
Series of positive real numbers. Measure theory.
*Journal of Formalized Mathematics*, 2, 1990. - [6]
Jozef Bialas.
The $\sigma$-additive measure theory.
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Jozef Bialas.
Several properties of the $\sigma$-additive measure.
*Journal of Formalized Mathematics*, 3, 1991. - [8]
Jozef Bialas.
Completeness of the $\sigma$-additive measure. Measure theory.
*Journal of Formalized Mathematics*, 4, 1992. - [9]
Jozef Bialas.
Some properties of the intervals.
*Journal of Formalized Mathematics*, 6, 1994. - [10]
Czeslaw Bylinski.
Basic functions and operations on functions.
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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.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama.
Basic properties of extended real numbers.
*Journal of Formalized Mathematics*, 12, 2000. - [15]
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama.
Definitions and basic properties of measurable functions.
*Journal of Formalized Mathematics*, 12, 2000. - [16]
Noboru Endou, Katsumi Wasaki, and Yasunari Shidama.
Some properties of extended real numbers operations: abs, min and max.
*Journal of Formalized Mathematics*, 12, 2000. - [17]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Andrzej Kondracki.
Basic properties of rational numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [19]
Andrzej Nedzusiak.
Probability.
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Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [21]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [22]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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