  Idempotent integration is an analogue of the Lebesgue integration where
$\sigma$-additive measures are replaced by $\sigma$-maxitive measures. It has
proved useful in many areas of mathematics such as fuzzy set theory,
optimization, idempotent analysis, large deviation theory, or extreme value
theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial
in all of these applications, was proved by Sugeno and Murofushi. Here we show
a converse statement to this idempotent version of the Radon--Nikodym theorem,
i.e. we characterize the $\sigma$-maxitive measures that have the
Radon--Nikodym property.
