On a characteristic of strongly embedded subspaces in symmetric spaces

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 $\eta_ {E}(H) <1. $ The last criterion is also valid for symmetric spaces “close” to $L_ {1},$ 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 $L_ {1},$ have the binary property: the characteristic $\eta_{E}(H)$ takes only two values, $0$ and $1$. This gives an example of binary Orlicz spaces different from the spaces $L_{p}$.