  This is a continuation of the papers [Kuryakov-Sukochev, JFA, 2015] and
[Sadovskaya-Sukochev, PAMS, 2018], in which the isomorphic classification of
$L_{p,q}$, for $1< p<\infty$, $1\le q<\infty$, $p\ne q $, on resonant measure
spaces, has been obtained. The aim of this paper is to give a complete
isomorphic classification of $L_{p,q}$-spaces on general $\sigma$-finite
measure spaces. Towards this end, several new subspaces of $L_{p,q}(0,1)$ and
$L_{p,q}(0,\infty)$ are identified and studied.
