The idempotent RadonNikodym theorem has a converse statement
Idempotent integration is an analog of the Lebesgue integration where ? ? -additive measures are replaced by ? ? -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 RadonNikodym derivatives, which turns out to be crucial in all these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the RadonNikodym theorem, i.e. we characterize the ? ? -maxitive measures that have the RadonNikodym property.
idempotent integration shilkret integral sugeno integral maxitive measure ?-principal measure pseudo-multiplication