Completeness of the $\sigma$-Additive Measure. Measure Theory

Jozef Bialas

University of Lodz

Summary.

Definitions and basic properties of a $\sigma$-additive, non-negative measure, with values in $\overline{\Bbb R}$, the enlarged set of real numbers, where $\overline{\Bbb R}$ denotes set $\overline{\Bbb R} = {\Bbb R}\cup\{-\infty,+\infty\}$ - by [10]. The article includes the text being a continuation of the paper [5]. Some theorems concerning basic properties of a $\sigma$-additive measure and completeness of the measure are proved.

MML Identifier: MEASURE3

Contents (PDF format)

Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.

[2] Jozef Bialas. Infimum and supremum of the set of real numbers. Measure theory. Journal of Formalized Mathematics, 2, 1990.

[3] Jozef Bialas. Series of positive real numbers. Measure theory. Journal of Formalized Mathematics, 2, 1990.

[4] Jozef Bialas. The $\sigma$-additive measure theory. Journal of Formalized Mathematics, 2, 1990.

[5] Jozef Bialas. Several properties of the $\sigma$-additive measure. Journal of Formalized Mathematics, 3, 1991.

[6] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[7] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.

[9] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.

[10] R. Sikorski. \em Rachunek rozniczkowy i calkowy - funkcje wielu zmiennych. Biblioteka Matematyczna. PWN - Warszawa, 1968.

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[12] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.

[13] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[14] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.