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 C of subsets of Ω, we find that the extension from C over the collection exp Ω of all subsets of Ω 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) 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.
A note on field-valued measures / Simone, Anna De; Hroch, Michal; Pták, Pavel. - In: MATHEMATICA SLOVACA. - ISSN 0139-9918. - 67:6(2017), pp. 1295-1300. [10.1515/ms-2017-0052]
A note on field-valued measures
Simone, Anna De;
2017
Abstract
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 C of subsets of Ω, we find that the extension from C over the collection exp Ω of all subsets of Ω 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) 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.