On weak convergence of filtrations

A sequence of filtrations (F(t)(n))tless than or equal toT converges weakly to a filtration (F(t))tless than or equal toT iff, for all B is an element of F(T), the sequence ofprocesses (E[1(B)\F(t)(n)])tless than or equal toT converges in probability under the Skorokhod topology to the process (E[1(B)\F(t)])tless than or equal toT. We give some examples of this kind of convergence; then we study, under the weak- convergence of liltrations, the convergence in probability of processes (E[X(t)\-F(t)(n)])tless than or equal toT to X where X is an F(t)-adapted semimartingale.