  In this paper we provide a systematic exposition of basic properties of
integrated distribution and quantile functions. We define these transforms in
such a way that they characterize any probability distribution on the real line
and are Fenchel conjugates of each other. We show that uniform integrability,
weak convergence and tightness admit a convenient characterization in terms of
integrated quantile functions. As an application we demonstrate how some basic
results of the theory of comparison of binary statistical experiments can be
deduced using integrated quantile functions. Finally, we extend the area of
application of the Chacon--Walsh construction in the Skorokhod embedding
problem.
