Fano's inequality for random variables

We extend fano's inequality, which controls the average probability of events in terms oh the average of some f-divergences, to work witj arbitrary events (not necessarily forming a partion) and even with arbitrary [0,1]-value random varibles, possibly on continously inifinete number. We provide two applications of these exstensions, in which the consideration of random varibles is particulary handy: we offer new and elegant proofs for existing lower bounds, on Bayesian posterior concentrations (minimax or distribution-dependent) rates and on the regret in nonstochastic sequential learning.