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

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.