  A one-to-one correspondence is drawn between law invariant risk measures and
divergences, which we define as functionals of pairs of probability measures on
arbitrary standard Borel spaces satisfying a few natural properties.
Divergences include many classical information divergence measures, such as
relative entropy and $f$-divergences. Several properties of divergence and
their duality with law invariant risk measures are developed, most notably
relating their chain rules or additivity properties with certain notions of
time consistency for dynamic law invariant risk measures known as acceptance
and rejection consistency. These properties are linked also to a peculiar
property of the acceptance sets on the level of distributions, analogous to
results of Weber on weak acceptance and rejection consistency. Finally, the
examples of shortfall risk measures and optimized certainty equivalents are
discussed in some detail, and it is shown that the relative entropy is
essentially the only divergence satisfying the chain rule.
