On Symmetric Spaces With Convergence in Measure on Reflexive Subspaces

Abstract

A closed subspace H of a symmetric space X on [0, 1] is said to be strongly embedded in X if in H the convergence in X-norm is equivalent to the convergence in measure. We study symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X. We prove that it is the case for all spaces, which satisfy an analogue of the classical Dunford–Pettis theorem on relatively weakly compact subsets in L1. At the same time the converse assertion fails for a broad class of separableMarcinkiewicz spaces.