Cameron-Martin theorem

The Cameron-Martin theorem, also called the Cameron-Martin formula, is a theorem of measure theory named after Robert Horton Cameron and W. T. Martin. It describes how abstract Wiener measure changes under translation by certain elements of the Cameron-Martin Hilbert space. The motivation is that standard Gaussian measure on n-dimensional Euclidean space is not translation-invariant, but under translation its Radon-Nikodym derivative with respect to the original measure is given by an explicit formula. The theorem shows that a similar formula holds for the abstract Wiener measure on a separable Banach space if one translates only by elements of a dense subspace. The restriction is essential: a locally finite Borel measure on a separable Banach space that is equivalent to its own push forward under every translation is either trivial or lives on a finite-dimensional space, so the formula cannot hold for arbitrary translations. The article also gives a general version for locally convex vector spaces.