Before revision:  In doing so we extend and unify a variety of "collapse to the mean" results from the literature and highlight the key role played by law invariance in causing the collapse.As a byproduct, we derive a number of applications to law-invariant acceptance sets and risk measures as well as Schur-convex functionals .
After revision:  In doing so we extend and unify a variety of "collapse to the mean" results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random variables. Our approach builds on two fundamental structural results for law-invariant acceptance sets and risk measures as well as Schur-convex functionals .