A note on field-valued measures
-
Anna De Simone
,
Michal Hroch
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.
The research of first author was partially supported by the PRIN research project “Metodi logici per il trattamento dell’informazione” (2010) of MIUR (Italian Ministry of University and Research). The second author was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS15/193/OHK3/3T/13
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© 2017 Mathematical Institute Slovak Academy of Sciences
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Artikel in diesem Heft
- Paolo de Lucia
- Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits
- A note on field-valued measures
- Topologies and uniformities on d0-algebras
- On some properties of 𝓙-approximately continuous functions
- Porous subsets in the space of functions having the Baire property
- Rademacher’s theorem in Banach spaces without RNP
- Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
- Measure games on pseudo-D-lattices
- On some properties of k-subadditive lattice group-valued capacities
- Lp Spaces in vector lattices and applications
- The Choquet integral with respect to fuzzy measures and applications
- A note on the range of vector measures
- Ideal convergent subsequences and rearrangements for divergent sequences of functions
- Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting
- A generalization of the exponential sampling series and its approximation properties
- Monotonicity and total boundedness in spaces of “measurable” functions
- Vector lattices in synaptic algebras
- Density, ψ-density and continuity
- Feather topologies
- On disruptions of nonautonomous discrete dynamical systems in the context of their local properties
- Rate of convergence of empirical measures for exchangeable sequences
- On non-additive probability measures
- Equi-topological entropy curves for skew tent maps in the square
- On the lack of equi-measurability for certain sets of Lebesgue-measurable functions
