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

## Infimum and Supremum of the Set of Real Numbers. Measure Theory

Jozef Bialas
University of Lodz

### Summary.

We introduce some properties of the least upper bound and the greatest lower bound of the subdomain of $\overline{\Bbb R}$ numbers, where $\overline{\Bbb R}$ denotes the enlarged set of real numbers, $\overline{\Bbb R} = {\Bbb R} \cup \{-\infty,+\infty\}$. The paper contains definitions of majorant and minorant elements, bounded from above, bounded from below and bounded sets, sup and inf of set, for nonempty subset of $\overline{\Bbb R}$. We prove theorems describing the basic relationships among those definitions. The work is the first part of the series of articles concerning the Lebesgue measure theory.

#### MML Identifier: SUPINF_1

The terminology and notation used in this paper have been introduced in the following articles [3] [2] [5] [1] [4]

Contents (PDF format)

#### Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[3] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[4] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[5] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

Received September 27, 1990

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