On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$
Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Restrepo Borrero, M. [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© Pleiades Publishing, Ltd. 2023 |
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| Übergeordnetes Werk: |
Enthalten in: P-adic numbers, ultrametric analysis, and applications - Moscow : MAIK Nauka, Interperiodica Publ., 2009, 15(2023), 1 vom: März, Seite 1-22 |
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| Übergeordnetes Werk: |
volume:15 ; year:2023 ; number:1 ; month:03 ; pages:1-22 |
| Links: |
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| DOI / URN: |
10.1134/S2070046623010016 |
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| Katalog-ID: |
SPR051559218 |
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| 100 | 1 | |a Restrepo Borrero, M. |e verfasserin |4 aut | |
| 245 | 1 | 0 | |a On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ |
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| 520 | |a Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. | ||
| 650 | 4 | |a Levi-Civita field |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a valued fields |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a ordered fields |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a non-Archimedean analysis |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a outer measure |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a measurable sets |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Srivastava, Vatsal |4 aut | |
| 700 | 1 | |a Shamseddine, K. |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t P-adic numbers, ultrametric analysis, and applications |d Moscow : MAIK Nauka, Interperiodica Publ., 2009 |g 15(2023), 1 vom: März, Seite 1-22 |w (DE-627)59571546X |w (DE-600)2487796-7 |x 2070-0474 |7 nnns |
| 773 | 1 | 8 | |g volume:15 |g year:2023 |g number:1 |g month:03 |g pages:1-22 |
| 856 | 4 | 0 | |u https://dx.doi.org/10.1134/S2070046623010016 |z lizenzpflichtig |3 Volltext |
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10.1134/S2070046623010016 doi (DE-627)SPR051559218 (SPR)S2070046623010016-e DE-627 ger DE-627 rakwb eng Restrepo Borrero, M. verfasserin aut On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2023 Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. Levi-Civita field (dpeaa)DE-He213 valued fields (dpeaa)DE-He213 ordered fields (dpeaa)DE-He213 non-Archimedean analysis (dpeaa)DE-He213 outer measure (dpeaa)DE-He213 measurable sets (dpeaa)DE-He213 Srivastava, Vatsal aut Shamseddine, K. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 15(2023), 1 vom: März, Seite 1-22 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:15 year:2023 number:1 month:03 pages:1-22 https://dx.doi.org/10.1134/S2070046623010016 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 15 2023 1 03 1-22 |
| spelling |
10.1134/S2070046623010016 doi (DE-627)SPR051559218 (SPR)S2070046623010016-e DE-627 ger DE-627 rakwb eng Restrepo Borrero, M. verfasserin aut On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2023 Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. Levi-Civita field (dpeaa)DE-He213 valued fields (dpeaa)DE-He213 ordered fields (dpeaa)DE-He213 non-Archimedean analysis (dpeaa)DE-He213 outer measure (dpeaa)DE-He213 measurable sets (dpeaa)DE-He213 Srivastava, Vatsal aut Shamseddine, K. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 15(2023), 1 vom: März, Seite 1-22 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:15 year:2023 number:1 month:03 pages:1-22 https://dx.doi.org/10.1134/S2070046623010016 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 15 2023 1 03 1-22 |
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10.1134/S2070046623010016 doi (DE-627)SPR051559218 (SPR)S2070046623010016-e DE-627 ger DE-627 rakwb eng Restrepo Borrero, M. verfasserin aut On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2023 Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. Levi-Civita field (dpeaa)DE-He213 valued fields (dpeaa)DE-He213 ordered fields (dpeaa)DE-He213 non-Archimedean analysis (dpeaa)DE-He213 outer measure (dpeaa)DE-He213 measurable sets (dpeaa)DE-He213 Srivastava, Vatsal aut Shamseddine, K. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 15(2023), 1 vom: März, Seite 1-22 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:15 year:2023 number:1 month:03 pages:1-22 https://dx.doi.org/10.1134/S2070046623010016 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 15 2023 1 03 1-22 |
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10.1134/S2070046623010016 doi (DE-627)SPR051559218 (SPR)S2070046623010016-e DE-627 ger DE-627 rakwb eng Restrepo Borrero, M. verfasserin aut On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2023 Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. Levi-Civita field (dpeaa)DE-He213 valued fields (dpeaa)DE-He213 ordered fields (dpeaa)DE-He213 non-Archimedean analysis (dpeaa)DE-He213 outer measure (dpeaa)DE-He213 measurable sets (dpeaa)DE-He213 Srivastava, Vatsal aut Shamseddine, K. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 15(2023), 1 vom: März, Seite 1-22 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:15 year:2023 number:1 month:03 pages:1-22 https://dx.doi.org/10.1134/S2070046623010016 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 15 2023 1 03 1-22 |
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10.1134/S2070046623010016 doi (DE-627)SPR051559218 (SPR)S2070046623010016-e DE-627 ger DE-627 rakwb eng Restrepo Borrero, M. verfasserin aut On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2023 Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. Levi-Civita field (dpeaa)DE-He213 valued fields (dpeaa)DE-He213 ordered fields (dpeaa)DE-He213 non-Archimedean analysis (dpeaa)DE-He213 outer measure (dpeaa)DE-He213 measurable sets (dpeaa)DE-He213 Srivastava, Vatsal aut Shamseddine, K. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 15(2023), 1 vom: März, Seite 1-22 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:15 year:2023 number:1 month:03 pages:1-22 https://dx.doi.org/10.1134/S2070046623010016 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 15 2023 1 03 1-22 |
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Enthalten in P-adic numbers, ultrametric analysis, and applications 15(2023), 1 vom: März, Seite 1-22 volume:15 year:2023 number:1 month:03 pages:1-22 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000naa a22002652 4500</leader><controlfield tag="001">SPR051559218</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230511064820.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">230511s2023 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1134/S2070046623010016</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR051559218</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)S2070046623010016-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Restrepo Borrero, M.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2023</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Pleiades Publishing, Ltd. 2023</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. 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Restrepo Borrero, M. |
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on a new measure on the levi-civita field %$ \mathcal{r} %$ |
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On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ |
| abstract |
Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. © Pleiades Publishing, Ltd. 2023 |
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Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. © Pleiades Publishing, Ltd. 2023 |
| abstract_unstemmed |
Abstract The Levi-Civita field %$ \mathcal{R} %$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on %$ \mathcal{R} %$ in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner and outer sums get closer together. That definition proved useful in developing an integration theory over %$ \mathcal{R} %$ in which the integral satisfies many of the essential properties of the Lebesgue integral of real analysis. Nevertheless, that measure theory lacks some intuitive results that one would expect in any reasonable definition for a measure; for example, the complement of a measurable set within another measurable set need not be measurable. In this paper, we will give a characterization for the measurable sets defined in [13]. Then we will introduce the notion of an outer measure on %$ \mathcal{R} %$ and show some key properties the outer measure has. Finally, we will use the notion of outer measure to define a new measure on %$ \mathcal{R} %$ that proves to be a better generalization of the Lebesgue measure from %$ \mathbb{R} %$ to %$ \mathcal{R} %$ and that leads to a family of measurable sets in %$ \mathcal{R} %$ that strictly contains the family of measurable sets from [13], and for which most of the classic results for Lebesgue measurable sets in %$ \mathbb{R} %$ hold. © Pleiades Publishing, Ltd. 2023 |
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On a New Measure on the Levi-Civita Field %$ \mathcal{R} %$ |
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Srivastava, Vatsal Shamseddine, K. |
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| score |
7.398368 |