# Purely non-atomic weak ${L}^{p}$ spaces

Studia Mathematica (1997)

- Volume: 122, Issue: 1, page 55-66
- ISSN: 0039-3223

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topLeung, Denny. "Purely non-atomic weak $L^p$ spaces." Studia Mathematica 122.1 (1997): 55-66. <http://eudml.org/doc/216360>.

@article{Leung1997,

abstract = {Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^\{p,∞\}(Ω,∑,μ)$ is isomorphic, as a Banach space, to $L^\{p,∞\}(Ω^\{\prime \},∑^\{\prime \},μ^\{\prime \})$ for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition $Ω = Ω_\{1\} ∪ Ω_\{2\}$ such that $(Ω_\{1\}, Σ ∩ Ω_\{1\},μ|_\{Σ ∩ Ω_\{1\}\})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $σ ⊆ Ω_\{2\}$. In particular, $L^\{p,∞\}(Ω,∑,μ)$ is isomorphic to $ℓ^\{p,∞\}$.},

author = {Leung, Denny},

journal = {Studia Mathematica},

keywords = {weak spaces; purely non-atomic measure space; measurable partition},

language = {eng},

number = {1},

pages = {55-66},

title = {Purely non-atomic weak $L^p$ spaces},

url = {http://eudml.org/doc/216360},

volume = {122},

year = {1997},

}

TY - JOUR

AU - Leung, Denny

TI - Purely non-atomic weak $L^p$ spaces

JO - Studia Mathematica

PY - 1997

VL - 122

IS - 1

SP - 55

EP - 66

AB - Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^{p,∞}(Ω,∑,μ)$ is isomorphic, as a Banach space, to $L^{p,∞}(Ω^{\prime },∑^{\prime },μ^{\prime })$ for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition $Ω = Ω_{1} ∪ Ω_{2}$ such that $(Ω_{1}, Σ ∩ Ω_{1},μ|_{Σ ∩ Ω_{1}})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $σ ⊆ Ω_{2}$. In particular, $L^{p,∞}(Ω,∑,μ)$ is isomorphic to $ℓ^{p,∞}$.

LA - eng

KW - weak spaces; purely non-atomic measure space; measurable partition

UR - http://eudml.org/doc/216360

ER -

## References

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- [3] D. H. Leung, Isomorphism of certain weak ${L}^{p}$ spaces, Studia Math. 104 (1993), 151-160. Zbl0814.46015
- [4] D. H. Leung, Isomorphic classification of atomic weak ${L}^{p}$ spaces, in: Interaction between Functional Analysis, Harmonic Analysis and Probability, N. J. Kalton, E. Saab and S. J. Montgomery-Smith (eds.), Marcel Dekker, 1996, 315-330. Zbl0836.46016
- [5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, 1977. Zbl0362.46013
- [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979. Zbl0403.46022
- [7] H. P. Lotz, Weak* convergence in the dual of weak ${L}^{p}$, unpublished manuscript. Zbl1201.46010
- [8] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974. Zbl0296.47023

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