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On two function spaces which are similar to $ L\sb 0$


Authors: S. J. Dilworth and D. A. Trautman
Journal: Proc. Amer. Math. Soc. 108 (1990), 451-456
MSC: Primary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1990-1000151-0
MathSciNet review: 1000151
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Abstract: Let $ {\Lambda _0}$ consist of all functions $ f$ measurable on $ \left( {0,\infty } \right)$ with

$\displaystyle \lambda \{ s:\vert f(s)\vert > t\} < \infty $

for all $ t > 0$, where $ \lambda $ is Lebesgue measure, and let $ {L_0}(0,\infty )$ consist of all measurable functions $ f$ with

$\displaystyle \mathop {\lim }\limits_{t \to \infty } \lambda \{ s:\vert f(s)\vert > t\} = 0.$

Let each space have the topology induced by convergence in measure. We show that every infinite-dimensional Banach subspace of $ {\Lambda _0}$ contains $ {c_0}$ or $ {l_p}$ for some $ 1 \leq p < \infty $. We also identify the duals of $ {\Lambda _0}$ and $ {L_0}(0,\infty )$.

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1990-1000151-0
Article copyright: © Copyright 1990 American Mathematical Society

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