<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In our previous article [22], we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a <italic>σ</italic>-field and the measure on it. In general, the existence of the measure on <italic>σ</italic>-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a <italic>σ</italic>-field. Therefore, in this article we define a measure (<monospace>pre-measure</monospace>) on a semialgebra and extend it to a measure on a <italic>σ</italic>-field. Furthermore, we give a <italic>σ</italic>-measure as an extension of the measure on a <italic>σ</italic>-field. We follow [24], [10], and [31].</p></abstract>

In our previous article [22], we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a σ-field and the measure on it. In general, the existence of the measure on σ-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a σ-field. Therefore, in this article we define a measure (pre-measure) on a semialgebra and extend it to a measure on a σ-field. Furthermore, we give a σ-measure as an extension of the measure on a σ-field. We follow [24], [10], and [31].

Construction of Measure from Semialgebra of Sets<xref ref-type="fn" rid="fn1_w2aab2b8b1b1b7b1ab1aaAa"><sup>1</sup></xref>

Gifu National College of Technology, Gifu, Japan

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Construction of Measure from Semialgebra of Sets

In our previous article [22], we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a...

Construction of Measure from Semialgebra of Sets¹

Publié en ligne: 25 mars 2016

Pages: 309 - 323

Reçu: 14 août 2015

DOI: https://doi.org/10.1515/forma-2015-0025

Mots clés
measure theory, pre-measure

© 2015 Noboru Endou, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Construction of Measure from Semialgebra of Sets1

Publié en ligne: 25 mars 2016

Pages: 309 - 323

Reçu: 14 août 2015

DOI: https://doi.org/10.1515/forma-2015-0025

Mots clésmeasure theory, pre-measure

© 2015 Noboru Endou, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Construction of Measure from Semialgebra of Sets¹

Mots clés
measure theory, pre-measure