  We extend Fano's inequality, which controls the average probability of events
in terms of the average of some $f$--divergences, to work with arbitrary events
(not necessarily forming a partition) and even with arbitrary $[0,1]$--valued
random variables, possibly in continuously infinite number. We provide two
applications of these extensions, in which the consideration of random
variables is particularly handy: we offer new and elegant proofs for existing
lower bounds, on Bayesian posterior concentration (minimax or
distribution-dependent) rates and on the regret in non-stochastic sequential
learning.
