We study operators
$$T:X \mapsto L_ \circ ([0,1],{\mathcal{M}},m)$$
(not necessarily linear) defined on a quasi-Bahach space *X* and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space
$$L_ \circ $$
are proved with the help of the Lorentz sequence spaces
$$l_{p,q} $$
. Sequences of functions belonging to fixed bounded sets in the spaces
$$L_{p,\infty } $$
are characterized for
$$0 < p < \infty $$
and
$$0 < q \leqslant p$$
. The possibility of distinguishing weak type operators (bounded in the space
$$L_{p,\infty } $$
) from operators factorizable through
$$L_{p,\infty } $$
is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in
$$L_{p,r} ,{\text{ }}0 < r \leqslant \infty $$
, is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.