Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
MathSciNet review: 1000151
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E30

Retrieve articles in all journals with MSC: 46E30


Additional Information

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