We will discuss the definition of coherent risk measures and its dual representation in this below. See its summary in [Rud07].

Let be a complete probability space. Let be a probability measure space, and . A function is called *coherent risk measure*, if it satisfies the following conditions:

- Normalization: ;
- Monotonicity: whenever ;
- Translation invariance: for any constant ;
- Subadditivity: ;
- Positive homogeneity: for any ;

Proposition 1is a weak* lower semicontinuous coherent risk measure if and only if there exists a non-empty subset of a probability measures with convex and weak* closed in , such that