On symmetric spaces with convergence in measure on reflexive subspaces

A closed subspace $H$ of a symmetric space $X$ on $[0,1]$ is said to be strongly embedded in $X$ if in $H$ a convergence in $X$-norm is equivalent to the convergence in Lebesgue 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 analog of the classical Dunford–Pettis theorem of relatively weakly compact subsets in $L_1$. At the same time the converse assertion fails for a wide class of separable Marcinkiewicz spaces.