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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2503.03793 |
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| _version_ | 1866913720341364736 |
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| author | Edalat, Abbas |
| author_facet | Edalat, Abbas |
| contents | We introduce the notion of a gauge and of a tagged partition (subordinate to a given gauge) by intersections of open and closed sets of a compact metric space extending the corresponding notions in Henstock-Kurzweil integration of real-valued functions with respect to the Lebesgue measure on the unit interval. We show that, for the integration of bounded functions with respect to a normalised Borel measure $μ$ on a compact metric space, the notion of a gauge and an associated tagged partition, arise naturally from a normalised simple valuation way-below the Borel measure. Then we consider the integration of unbounded functions with respect to a normalised Borel measure on a compact metric space, for which the Lebesgue integral may fail to exist. A pair of a tagged partition and a gauge defines a simple valuation and we introduce a partial order on these pairs, emulating the partial order of simple valuations in the probabilistic power domain. We define the $D_μ$-integral of a real-valued function with respect to a Borel measure using the limit of the net of the integrals of the simple valuations induced by pairs of tagged partitions and gauges for the function. The $D_μ$-integral of functions on a compact metric space with respect to a normalised Borel measure satisfies the basic properties of an integral and generalises the Henstock-Kurzweil integral. We show that when the Lebesgue integral of the function exists then the $D_μ$-integral also exists and they have the same value. We provide a family of real-valued functions on the Cantor space that are $D_μ$-integrable but not Lebesgue integrable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A generalisation of Henstock-Kurzweil integral to compact metric spaces Edalat, Abbas Functional Analysis 28C05 We introduce the notion of a gauge and of a tagged partition (subordinate to a given gauge) by intersections of open and closed sets of a compact metric space extending the corresponding notions in Henstock-Kurzweil integration of real-valued functions with respect to the Lebesgue measure on the unit interval. We show that, for the integration of bounded functions with respect to a normalised Borel measure $μ$ on a compact metric space, the notion of a gauge and an associated tagged partition, arise naturally from a normalised simple valuation way-below the Borel measure. Then we consider the integration of unbounded functions with respect to a normalised Borel measure on a compact metric space, for which the Lebesgue integral may fail to exist. A pair of a tagged partition and a gauge defines a simple valuation and we introduce a partial order on these pairs, emulating the partial order of simple valuations in the probabilistic power domain. We define the $D_μ$-integral of a real-valued function with respect to a Borel measure using the limit of the net of the integrals of the simple valuations induced by pairs of tagged partitions and gauges for the function. The $D_μ$-integral of functions on a compact metric space with respect to a normalised Borel measure satisfies the basic properties of an integral and generalises the Henstock-Kurzweil integral. We show that when the Lebesgue integral of the function exists then the $D_μ$-integral also exists and they have the same value. We provide a family of real-valued functions on the Cantor space that are $D_μ$-integrable but not Lebesgue integrable. |
| title | A generalisation of Henstock-Kurzweil integral to compact metric spaces |
| topic | Functional Analysis 28C05 |
| url | https://arxiv.org/abs/2503.03793 |