Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991
Association of Mizar Users
Several Properties of the $\sigma$additive Measure

Jozef Bialas

University of Lodz
Summary.

A continuation of [4].
The paper contains the definition and basic properties of a
$\sigma$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 R.~Sikorski [9]. Some simple theorems concerning basic
properties of a $\sigma$additive measure,
measurable sets, measure zero sets are proved. The work is the fourth 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
[10]
[7]
[12]
[11]
[13]
[5]
[6]
[1]
[8]
[2]
[3]
[4]
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]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
 [6]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
 [7]
Czeslaw Bylinski.
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Journal of Formalized Mathematics,
1, 1989.
 [8]
Beata Padlewska.
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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.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
 [11]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
 [12]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
 [13]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received July 3, 1991
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