On two function spaces which are similar to $L_ 0$

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 ${L_0}(0,\infty )$.