A note on field-valued measures

Rok, strany: 2017, 1295 - 1300

We consider the Horn-Tarski condition for the extension of (signed) measures (resp., non-negative measures) in the setup of field-valued assignments. For a finite collection $\mathcal{C}$ of subsets of $\Omega$, we find that the extension from $\mathcal{C}$ over the collection $\exp\Omega$ of all subsets of $\Omega$ is implied by, and indeed equivalent to, a certain type of Frobenius theorem (resp. a certain type of Farkas lemma). This links classical notions of linear algebra with a generalized version of Horn-Tarski condition on extensions of measures. We also observe that for a general (infinite) $\mathcal{C}$ the Horn-Tarski condition guarantees the extension of signed measures (here the standard Zorn lemma applies). However, we find out that the extensions for non-negative ordered-field-valued measures are generally not available.

ISO 690:

De Simone, A., Hroch, M., Pták, P. 2017. A note on field-valued measures. In Mathematica Slovaca, vol. 67, no.6, pp. 1295-1300. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0052

APA:

De Simone, A., Hroch, M., Pták, P. (2017). A note on field-valued measures. Mathematica Slovaca, 67(6), 1295-1300. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0052