  We characterize when a convex risk measure associated to a law-invariant
acceptance set in $L^\infty$ can be extended to $L^p$, $1\leq p<\infty$,
preserving finiteness and continuity. This problem is strongly connected to the
statistical robustness of the corresponding risk measures. Special attention is
paid to concrete examples including risk measures based on expected utility,
max-correlation risk measures, and distortion risk measures.
