A Probabilistic Generalization of the Mazur-Ulam Theorem

TL;DR

This paper extends the classical Mazur-Ulam theorem to probabilistic settings, showing that maps preserving distances almost everywhere are essentially Euclidean isometries, broadening its applicability in applied mathematics.

Contribution

It provides a rigorous probabilistic generalization of the Mazur-Ulam theorem, characterizing almost everywhere distance-preserving maps as Euclidean isometries.

Findings

Maps preserving distances almost everywhere are almost surely Euclidean isometries.

The result applies to measurable maps on subsets of Rd with full-dimensional support.

The theorem bridges classical isometry results with probabilistic and measure-theoretic contexts.

Abstract

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.