  We study a space of coherent risk measures M_phi obtained as certain
expansions of coherent elementary basis measures. In this space, the concept of
``Risk Aversion Function'' phi naturally arises as the spectral representation
of each risk measure in a space of functions of confidence level probabilities.
We give necessary and sufficient conditions on phi for M_phi to be a coherent
measure. We find in this way a simple interpretation of the concept of
coherence and a way to map any rational investor's subjective risk aversion
onto a coherent measure and vice--versa. We also provide for these measures
their discrete versions M_phi^N acting on finite sets of N independent
realizations of a r.v. which are not only shown to be coherent measures for any
fixed N, but also consistent estimators of M_phi for large N. Finally, we find
in our results some interesting and not yet fully investigated relationships
with certain results known in insurance mathematical literature.
