In the classical theory of monotone equimeasurable rearrangements of functions, “equimeasurability” (i.e., that two functions have the same distribution) is defined relative to a given additive probability measure. These rearrangement tools have been successfully used in many problems in economic theory dealing with uncertainty where the monotonicity of a solution is desired. However, in all of these problems, uncertainty refers to the classical Bayesian understanding of the term, where the idea of ambiguity is absent. Arguably, Knightian uncertainty, or ambiguity, is one of the cornerstones of modern decision theory. It is hence natural to seek an extension of these classical tools of equimeasurable rearrangements to the non-Bayesian or neo-Bayesian context. This paper introduces the idea of a monotone equimeasurable rearrangement in the context of nonadditive probabilities, or capacities that satisfy a property that I call strong diffuseness. The latter is a strengthening of the usual notion of diffuseness, and these two properties coincide for additive measures and for submodular (i.e., concave) capacities. To illustrate the usefulness of these tools in economic theory, I consider an application to a problem arising in the theory of production under uncertainty.