Tensor valuations on Sobolev spaces

A complete classification is established for continuous, SL($n$) contravariant, and translation invariant tensor valuations defined on the Sobolev space $W^{1,p}({\mathbb{R}}^n)$. When these valuations are further assumed to be homogeneous, the classification reveals that they are precisely the Fisher information tensors, which constitute a higher-order generalization of the Fisher information matrix.