The Measurability of Extended Real Valued Functions

Noboru Endou

Shinshu University, Nagano

Katsumi Wasaki

Shinshu University, Nagano

Yasunari Shidama

Shinshu University, Nagano

Summary.

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.

MML Identifier: MESFUNC2

Contents (PDF format)

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

Bibliography

[1] 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. Journal of Formalized Mathematics, 2, 1990.

[7] 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. Journal of Formalized Mathematics, 1, 1989.

[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. 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. Journal of Formalized Mathematics, 2, 1990.

[20] 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.