Before revision:  The same approach can be also successfully employed to obtain an elementary proof of the dual representation of "first aggregate , then allocate" -type systemic risk measures. As a final application, we apply our results to derive a simple proof of the dual representation of univariate utility-based risk measures  .
After revision:  The same approach can be also successfully employed to obtain an elementary proof of the dual representation of "first aggregate , then allocate" type. In both cases, we provide a detailed analysis of the corresponding systemic acceptance sets and their support functions. The same approach delivers a simple and self-contained proof of the dual representation of univariate utility-based risk measures .