Abstract
It is shown that the presence of a lower p - estimate with constant 1 in the symmetric space E is sufficient for the condition of equivalence of convergence in norm and in measure on the subspace H of the space E to be satisfied if and only if the numerical characteristic E(H) 1. The last criterion is also valid for symmetric spaces close to L1, more precisely, for which an analog of the Dunford - Pettis criterion of weak compactness is valid. In particular, it is shown that spaces close to L1, have the binary property: the characteristic E(H) takes only two values, 0 and 1. This gives an example of binary Orlicz spaces differentfrom the spaces Lp.
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