We study combinations of risk measures under no restrictive assumption on the set of alternatives. The main result is the representation for resulting risk measures from the properties of both alternative functionals and combination functions. To that, we develop a representation for arbitrary mixture of convex risk measures. In this case, we obtain a penalty that recall the notion of inf-convolution under theoretical measure integration. As an application, we address the context of probability-based risk measurements for functionals on the set of distribution functions. We develop results related to this specific context. We also explore features of individual interest generated by our framework, such as the preservation of continuity properties, the representation of worst-case risk measures, stochastic dominance and elicitability.
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