  We establish a profound connection between coherent risk measures, a
prominent object in quantitative finance, and uniform integrability, a
fundamental concept in probability theory. Instead of working with absolute
values of random variables, which is convenient in studying integrability, we
work directly with random loses and gains, which have clear financial
interpretation. We introduce a technical tool called the folding score of
distortion risk measures. The analysis of the folding score allows us to
convert some conditions on absolute values to those on gains and losses. As our
main results, we obtain three sets of equivalent conditions for uniform
integrability. In particular, a set is uniformly integrable if and only if one
can find a coherent distortion risk measure that is bounded on the set, but not
finite on $L^1$.
