The information topology

If (Q_n)_n∈N is a sequence of probability distributions, we say that (Q_n)_n∈N converges to Q in information, if D(Qn||Q)→0 for n→∞, where D(P||Q)=∫ log(dP/dQ)dP is the information divergence from P to Q. Convergence in information has been proved in many cases of statistical importance. On the set of probability distributions M₊¹ (Ω,B) there exists several relevant topologies. For instance the strong topology is defined by the total variation norm ||P-Q||=sup_|f|≤1{∫_ΩfdP-∫_ΩfdQ}. Since it has long been known that the divergence balls B(Q, r)={P∈M₊¹ (Ω,B) | D(P || Q)\n\n\t\t