  This paper offers a mathematical invention that shows how to convert
integrated quantiles, which often appear in risk measures, into integrated
cumulative distribution functions, which are technically more tractable from
various perspectives. The invention helps to avoid a number of technical
assumptions that have been traditionally imposed when working with quantities
containing quantiles. In particular it helps to completely avoid the
requirement of the existence of a probability density function. The developed
results explain and illustrate the invention, whose byproducts include the
assessment of model uncertainty and misspecification, and the derivation of
statistical inference results.
