  We unify standard frameworks for approachability both in full or partial
monitoring by defining a new abstract game, called the "purely informative
game", where the outcome at each stage is the maximal information players can
obtain, represented as some probability measure. Objectives of players can be
rewritten as the convergence (to some given set) of sequences of averages of
these probability measures. We obtain new results extending the approachability
theory developed by Blackwell moreover this new abstract framework enables us
to characterize approachable sets with, as usual, a remarkably simple and clear
reformulation for convex sets. Translated into the original games, those
results become the first necessary and sufficient condition under which an
arbitrary set is approachable and they cover and extend previous known results
for convex sets. We also investigate a specific class of games where, thanks to
some unusual definition of averages and convexity, we again obtain a complete
characterization of approachable sets along with rates of convergence.
