On two function spaces which are similar to $L_ 0$
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- by S. J. Dilworth and D. A. Trautman PDF
- Proc. Amer. Math. Soc. 108 (1990), 451-456 Request permission
Abstract:
Let ${\Lambda _0}$ consist of all functions $f$ measurable on $\left ( {0,\infty } \right )$ with \[ \lambda \{ s:|f(s)| > t\} < \infty \] for all $t > 0$, where $\lambda$ is Lebesgue measure, and let ${L_0}(0,\infty )$ consist of all measurable functions $f$ with \[ \lim \limits _{t \to \infty } \lambda \{ s:|f(s)| > t\} = 0.\] 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
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Additional Information
- © Copyright 1990 American Mathematical Society
- 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