It is well known that the only locally finite, translation-invariant Borel measure on an
infinite-dimensional, separable Frechet space is the trivial measure.  I am wondering about an analogous proposition for Frechet manifolds (e.g., the manifold of Lorentz metrics over a candidate spacetime manifold, a 4-dimensional, Hausdorff, smooth manifold).  Define the notion of "locally translation-invariant measure" as follows:  fix a point p of the Frechet manifold, a chart (O, \phi) containing the point and a measurable neighborhood N of p contained in O; then any translation of N using the local Frechet linear structure that leaves N entirely in O preserves the measure of N.  Then I think the following is likely true:  There is no locally finite, locally translation-invariant Borel measure on an infinite-dimensional, separable Frechet manifold.  The proof would use probably use the fact that every infinite-dimensional, separable Frechet manifold is isomorphic to an open subset of the infinite-dimensional, separable Hilbert space (on which there is, of course, no such measure).  Is this result, or something close to it, or a counter-example, known?

Thanks!

Erik Curiel