  We introduce the concept of set risk measures (SRMs), which are real-valued
maps defined on the space of all non-empty, closed, and bounded sets of almost
surely bounded random variables. Traditional risk measures typically operate on
spaces of random variables, but SRMs extend this framework to sets of random
variables. We establish an axiom scheme for SRMs, similar to classical risk
measures but adapted for set operations. The main technical contribution is an
axiomatic dual representation of convex SRMs by using regular, finitely
additive measures on the unit ball of the dual space of essentially bounded
random variables. We explore worst-case SRMs, which evaluate risk as the
supremum of individual risks within a set, and provide a collection of examples
illustrating the applicability of our framework to systemic risk, portfolio
optimization, and decision-making under uncertainty. This work extends the
theory of risk measures to a more general and flexible setup, accommodating a
broader range of financial and mathematical applications.
