Finite representability of $\ell_p$-spaces in symmetric spaces

For a separable rearrangement invariant space $X$ on the semiaxis, ${\mathcal F}(X)$ is defined to be the set of all $p\in [1,\infty ]$ such that $\ell _p$ is finitely representable in $X$ in such a way that the standard basis vectors of $\ell _p$ correspond to equimeasurable mutually disjoint functions. In the paper, a characterization of the set ${\mathcal F}(X)$ is obtained. As a consequence, a version of Krivine's well-known theorem is proved for rearrangement invariant spaces. Next, a description of the sets ${\mathcal F}(X)$ for certain Lorentz spaces is found.