Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## The $\sigma$-additive Measure Theory

Jozef Bialas
University of Lodz

### Summary.

The article contains definition and basic properties of $\sigma$-additive, nonnegative 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 [9]. We present definitions of $\sigma$-field of sets, $\sigma$-additive measure, measurable sets, measure zero sets and the basic theorems describing relationships between the notion mentioned above. The work is the third part of the series of articles concerning the Lebesgue measure theory.

#### MML Identifier: MEASURE1

The terminology and notation used in this paper have been introduced in the following articles [11] [10] [5] [14] [12] [15] [13] [3] [4] [8] [6] [7] [1] [2]

Contents (PDF format)

#### Bibliography

[1] Jozef Bialas. Infimum and supremum of the set of real numbers. Measure theory. Journal of Formalized Mathematics, 2, 1990.
[2] Jozef Bialas. Series of positive real numbers. Measure theory. Journal of Formalized Mathematics, 2, 1990.
[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[6] Andrzej Nedzusiak. $\sigma$-fields and probability. Journal of Formalized Mathematics, 1, 1989.
[7] Andrzej Nedzusiak. Probability. Journal of Formalized Mathematics, 2, 1990.
[8] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[9] R. Sikorski. \em Rachunek rozniczkowy i calkowy - funkcje wielu zmiennych. Biblioteka Matematyczna. PWN - Warszawa, 1968.
[10] Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.
[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] Andrzej Trybulec and Agata Darmochwal. Boolean domains. Journal of Formalized Mathematics, 1, 1989.
[14] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[15] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.