If X is a Banach space that is isomorphic to a closed subspace of L_0 (the space of measurable functions equipped with convergence in measure), must X be isomorphic to a closed subspace of L_1? Assume Lebesgue measurable functions on the unit interval. N. Kalton showed that X is isomorphic to linear subspace of L_1 if and only if little-L_1(X) is isomorphic to a closed subspace of L_0.
What are the known consequences of an answer to this question, positive or negative?
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