A class of weakly compact sets in Lebesgue–Bochner spaces
Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, con...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Rodríguez, José [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017transfer abstract |
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| Schlagwörter: |
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| Umfang: |
13 |
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| Übergeordnetes Werk: |
Enthalten in: Frequent mutations in the RPL22 gene and its clinical and functional implications - 2013, a journal devoted to general, geometric, set-theoretic and algebraic topology, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:222 ; year:2017 ; day:15 ; month:05 ; pages:16-28 ; extent:13 |
| Links: |
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| DOI / URN: |
10.1016/j.topol.2017.02.075 |
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| 520 | |a Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . | ||
| 520 | |a Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . | ||
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10.1016/j.topol.2017.02.075 doi GBVA2017005000011.pica (DE-627)ELV035730714 (ELSEVIER)S0166-8641(17)30154-2 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 610 VZ 44.11 bkl Rodríguez, José verfasserin aut A class of weakly compact sets in Lebesgue–Bochner spaces 2017transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . 46G10 Elsevier 46B50 Elsevier Enthalten in Elsevier Frequent mutations in the RPL22 gene and its clinical and functional implications 2013 a journal devoted to general, geometric, set-theoretic and algebraic topology Amsterdam [u.a.] (DE-627)ELV011305770 volume:222 year:2017 day:15 month:05 pages:16-28 extent:13 https://doi.org/10.1016/j.topol.2017.02.075 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_22 GBV_ILN_40 GBV_ILN_62 GBV_ILN_73 GBV_ILN_252 44.11 Präventivmedizin VZ AR 222 2017 15 0515 16-28 13 045F 510 |
| spelling |
10.1016/j.topol.2017.02.075 doi GBVA2017005000011.pica (DE-627)ELV035730714 (ELSEVIER)S0166-8641(17)30154-2 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 610 VZ 44.11 bkl Rodríguez, José verfasserin aut A class of weakly compact sets in Lebesgue–Bochner spaces 2017transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . 46G10 Elsevier 46B50 Elsevier Enthalten in Elsevier Frequent mutations in the RPL22 gene and its clinical and functional implications 2013 a journal devoted to general, geometric, set-theoretic and algebraic topology Amsterdam [u.a.] (DE-627)ELV011305770 volume:222 year:2017 day:15 month:05 pages:16-28 extent:13 https://doi.org/10.1016/j.topol.2017.02.075 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_22 GBV_ILN_40 GBV_ILN_62 GBV_ILN_73 GBV_ILN_252 44.11 Präventivmedizin VZ AR 222 2017 15 0515 16-28 13 045F 510 |
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10.1016/j.topol.2017.02.075 doi GBVA2017005000011.pica (DE-627)ELV035730714 (ELSEVIER)S0166-8641(17)30154-2 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 610 VZ 44.11 bkl Rodríguez, José verfasserin aut A class of weakly compact sets in Lebesgue–Bochner spaces 2017transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . 46G10 Elsevier 46B50 Elsevier Enthalten in Elsevier Frequent mutations in the RPL22 gene and its clinical and functional implications 2013 a journal devoted to general, geometric, set-theoretic and algebraic topology Amsterdam [u.a.] (DE-627)ELV011305770 volume:222 year:2017 day:15 month:05 pages:16-28 extent:13 https://doi.org/10.1016/j.topol.2017.02.075 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_22 GBV_ILN_40 GBV_ILN_62 GBV_ILN_73 GBV_ILN_252 44.11 Präventivmedizin VZ AR 222 2017 15 0515 16-28 13 045F 510 |
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10.1016/j.topol.2017.02.075 doi GBVA2017005000011.pica (DE-627)ELV035730714 (ELSEVIER)S0166-8641(17)30154-2 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 610 VZ 44.11 bkl Rodríguez, José verfasserin aut A class of weakly compact sets in Lebesgue–Bochner spaces 2017transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . 46G10 Elsevier 46B50 Elsevier Enthalten in Elsevier Frequent mutations in the RPL22 gene and its clinical and functional implications 2013 a journal devoted to general, geometric, set-theoretic and algebraic topology Amsterdam [u.a.] (DE-627)ELV011305770 volume:222 year:2017 day:15 month:05 pages:16-28 extent:13 https://doi.org/10.1016/j.topol.2017.02.075 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_22 GBV_ILN_40 GBV_ILN_62 GBV_ILN_73 GBV_ILN_252 44.11 Präventivmedizin VZ AR 222 2017 15 0515 16-28 13 045F 510 |
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10.1016/j.topol.2017.02.075 doi GBVA2017005000011.pica (DE-627)ELV035730714 (ELSEVIER)S0166-8641(17)30154-2 DE-627 ger DE-627 rakwb eng 510 510 DE-600 610 VZ 610 VZ 44.11 bkl Rodríguez, José verfasserin aut A class of weakly compact sets in Lebesgue–Bochner spaces 2017transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . 46G10 Elsevier 46B50 Elsevier Enthalten in Elsevier Frequent mutations in the RPL22 gene and its clinical and functional implications 2013 a journal devoted to general, geometric, set-theoretic and algebraic topology Amsterdam [u.a.] (DE-627)ELV011305770 volume:222 year:2017 day:15 month:05 pages:16-28 extent:13 https://doi.org/10.1016/j.topol.2017.02.075 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_22 GBV_ILN_40 GBV_ILN_62 GBV_ILN_73 GBV_ILN_252 44.11 Präventivmedizin VZ AR 222 2017 15 0515 16-28 13 045F 510 |
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A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. 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a class of weakly compact sets in lebesgue–bochner spaces |
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A class of weakly compact sets in Lebesgue–Bochner spaces |
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Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . |
| abstractGer |
Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . |
| abstract_unstemmed |
Let X be a Banach space and μ a probability measure. A set K ⊆ L 1 ( μ , X ) is said to be a δ S -set if it is uniformly integrable and for every δ > 0 there is a weakly compact set W ⊆ X such that μ ( f − 1 ( W ) ) ≥ 1 − δ for every f ∈ K . This is a sufficient, but in general non-necessary, condition for relative weak compactness in L 1 ( μ , X ) . We say that X has property ( δ S μ ) if every relatively weakly compact subset of L 1 ( μ , X ) is a δ S -set. In this paper we study δ S -sets and Banach spaces having property ( δ S μ ). We show that testing on uniformly bounded sets is enough to check this property. New examples of spaces having property ( δ S μ ) are provided. Special attention is paid to the relationship with strongly weakly compactly generated (SWCG) spaces. In particular, we show an example of a SWCG (in fact, separable Schur) space failing property ( δ S μ ) when μ is the Lebesgue measure on [ 0 , 1 ] . |
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A class of weakly compact sets in Lebesgue–Bochner spaces |
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