On unconditional convergence in the space $L_1$

The paper contains a proof of the following
Theorem. {\it Suppose $\sum_{k=1}^\infty f_k(x)$ converges unconditionally in $L_1[0,1]$. Then for any $\varepsilon>0$ there exists a set $E_\varepsilon\subset[0,1],$ $\mu E_\varepsilon>1-\varepsilon,$ such that $\sum_{k=1}^\infty f_k(x)$ converges unconditionally in $L_q(E_\varepsilon)$ for every $q<2$.}
This result is obtained as a corollary of a more general theorem.
Bibliography: 2 titles.