  We study the predictable representation property in the progressive
enlargement F^\tau of a reference filtration F by a random time \tau. Our
approach is based on the decomposition of any random time into two parts, one
overlapping F-stopping times (thin part) and the other one that avoids
F-stopping times (thick part). We assume that the F-thin part of \tau is
nontrivial and prove a martingale representation theorem on F^\tau. We thus
extend previous results dealing with F-avoiding random times. We collect some
examples of application to the enlargement of the natural filtration of a
L\'evy process.
