Abstract
The paper contains a gap in the justification of the uniform non degeneracy of the measures , , asserted in Proposition 2.8.1; we used a 0-1 law argument which cannot hold in general due to a conditioning problem; we provide an alternative argument. In the terminology of branching processes, this argument yields new sufficient conditions for an additive martingale on a Galton-Watson tree in varying environment to be almost surely positive conditionally on non extinction of the tree.