# Hopf Extension Theorem of Measure

Noboru Endou; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2009)

- Volume: 17, Issue: 2, page 157-162
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topNoboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. "Hopf Extension Theorem of Measure." Formalized Mathematics 17.2 (2009): 157-162. <http://eudml.org/doc/266742>.

@article{NoboruEndou2009,

abstract = {The authors have presented some articles about Lebesgue type integration theory. In our previous articles [12, 13, 26], we assumed that some σ-additive measure existed and that a function was measurable on that measure. However the existence of such a measure is not trivial. In general, because the construction of a finite additive measure is comparatively easy, to induce a σ-additive measure a finite additive measure is used. This is known as an E. Hopf's extension theorem of measure [15].},

author = {Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama},

journal = {Formalized Mathematics},

language = {eng},

number = {2},

pages = {157-162},

title = {Hopf Extension Theorem of Measure},

url = {http://eudml.org/doc/266742},

volume = {17},

year = {2009},

}

TY - JOUR

AU - Noboru Endou

AU - Hiroyuki Okazaki

AU - Yasunari Shidama

TI - Hopf Extension Theorem of Measure

JO - Formalized Mathematics

PY - 2009

VL - 17

IS - 2

SP - 157

EP - 162

AB - The authors have presented some articles about Lebesgue type integration theory. In our previous articles [12, 13, 26], we assumed that some σ-additive measure existed and that a function was measurable on that measure. However the existence of such a measure is not trivial. In general, because the construction of a finite additive measure is comparatively easy, to induce a σ-additive measure a finite additive measure is used. This is known as an E. Hopf's extension theorem of measure [15].

LA - eng

UR - http://eudml.org/doc/266742

ER -

## References

top- [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [3] Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.
- [4] Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.
- [5] Józef Białas. Several properties of the σ-additive measure. Formalized Mathematics, 2(4):493-497, 1991.
- [6] Józef Białas. The σ-additive measure theory. Formalized Mathematics, 2(2):263-270, 1991.
- [7] Józef Białas. Properties of Caratheodor's measure. Formalized Mathematics, 3(1):67-70, 1992.
- [8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [9] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [10] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [11] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [12] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008, doi:10.2478/v10037-008-0023-1.[Crossref] Zbl1321.46022
- [13] Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006, doi:10.2478/v10037-006-0008-x.[Crossref]
- [14] Adam Grabowski. On the Kuratowski limit operators. Formalized Mathematics, 11(4):399-409, 2003.
- [15] P. R. Halmos. Measure Theory. Springer-Verlag, 1987.
- [16] Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.
- [17] Franz Merkl. Dynkin's lemma in measure theory. Formalized Mathematics, 9(3):591-595, 2001.
- [18] Andrzej Nędzusiak. Probability. Formalized Mathematics, 1(4):745-749, 1990.
- [19] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.
- [20] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
- [21] Karol Pąk. The Nagata-Smirnov theorem. Part II. Formalized Mathematics, 12(3):385-389, 2004.
- [22] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
- [23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [24] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [25] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
- [26] Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007, doi:10.2478/v10037-007-0026-3.[Crossref]
- [27] Bo Zhang, Hiroshi Yamazaki, and Yatsuka Nakamura. Set sequences and monotone class. Formalized Mathematics, 13(4):435-441, 2005.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.