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

Studia Mathematica (1997)

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

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## Abstract

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Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If ${L}^{p,\infty }\left(\Omega ,\sum ,\mu \right)$ is isomorphic, as a Banach space, to ${L}^{p,\infty }\left({\Omega }^{\text{'}},{\sum }^{\text{'}},{\mu }^{\text{'}}\right)$ for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition $\Omega ={\Omega }_{1}\cup {\Omega }_{2}$ such that $\left({\Omega }_{1},\Sigma \cap {\Omega }_{1},\mu {|}_{\Sigma \cap {\Omega }_{1}}\right)$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $\sigma \subseteq {\Omega }_{2}$. In particular, ${L}^{p,\infty }\left(\Omega ,\sum ,\mu \right)$ is isomorphic to ${\ell }^{p,\infty }$.

## How to cite

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Leung, 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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1. [1] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces ${L}_{p}$, Studia Math. 21 (1962), 161-176.
2. [2] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, 1974. Zbl0285.46024
3. [3] D. H. Leung, Isomorphism of certain weak ${L}^{p}$ spaces, Studia Math. 104 (1993), 151-160. Zbl0814.46015
4. [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. [5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, 1977. Zbl0362.46013
6. [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979. Zbl0403.46022
7. [7] H. P. Lotz, Weak* convergence in the dual of weak ${L}^{p}$, unpublished manuscript. Zbl1201.46010
8. [8] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974. Zbl0296.47023

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