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.
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.
Received October 15, 1990
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