A Probabilistic Generalization of the Mazur-Ulam Theorem

The classical Mazur-Ulam theorem establishes that every surjective isometry between normed real vector spaces is an affine transformation. In various applied mathematical settings, however, one encounters maps that preserve distances not pointwise, but almost everywhere with respect to a probability measure. This paper provides a rigorous generalization of the Mazur-Ulam theorem to probability spaces. We prove that if a measurable map on a subset of Rd preserves distances almost everywhere with respect to a measure with full-dimensional support, it coincides almost everywhere with a global Euclidean isometry, defined as an orthogonal transformation followed by a translation.