Let $𝒳\to \Delta$ a smooth projective family and $(L,h)$ a pseudo-effective line bundle on $𝒳$ (i.e. with a non-negative curvature current $\Theta hL$). In its works on invariance of plurigenera, Y.-T. Siu was interested in extending sections of $m{K}_{{𝒳}_0}+L$ (defined over the central fiber of the family ${𝒳}_0$) to sections of $m{K}_{𝒳}+L$. In this article we consider the following problem: to extend sections of $m(K_{𝒳}+L)$. More precisely, we show the following result: assuming the triviality of the multiplier ideal sheaf $ℐ({𝒳}_0,h_{|{𝒳}_0})$, any section of $m(K_{{𝒳}_0}+L)$ extends to $𝒳$ ; in other...