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On integration with respect to a DT-measure

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Abstract

DT-measures generalize the topological measures of Aarnes. We continue to study an integral of a real-valued continuous function on compact Hausdorff space with respect to a DT-measure and investigate the following: multiplicativity of this integral, \(w*\)-topology on the family of DT-measures and density of various subfamilies of this space, and also integral as a set function.

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Notes

  1. For a T-measure this theorem was obtained in [7].

  2. In [11] integral (3.1) is called r-integral in contrast to l-integral, which is not considered in this paper. If \(\psi \in TM\), then r- and l-integrals coinside with the integral from [1].

  3. Let T be a topological space and \(Y,Z \subset T\). We say that Y is dense in Z if \(Z \subset \overline{Y}\).

  4. Note that any \(X_i\) be a compact Hausdorff space with the topology induced from X.

  5. A signed T-measure is a set function from \(\alpha \) to \(\mathbb {R}\) with properties (m1)–(m4), which besides is additive on \(\alpha \) as whole. It was considered for the first time in [6] under the name of signed quasi-measure.

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Correspondence to Marina Svistula.

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Svistula, M. On integration with respect to a DT-measure. Positivity 20, 579–598 (2016). https://doi.org/10.1007/s11117-015-0373-1

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  • DOI: https://doi.org/10.1007/s11117-015-0373-1

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