The space $L_1(L_p)$ is primary for $1<p<infty$

The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$|f| = int_0^1Big(int_0^1 |f(x,y)|^p dyBig)^{1/p}dx < infty.$$ We show that $L_1(L_p)$ $(1 < p < infty)$ is primary, meaning that, whenever $L_1(L_p) = Eoplus F$ then either $E$ or $F$ is isomorphic to $L_1(L_p)$. More generally we show that $L_1(X)$ is primary, for a large class of rearrangement invariant Banach function spaces.