It is well known that there are no Lebesgue measures on infinite-dimensional Banach spaces (see e.g. http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure). However, I couldn't find anything about Lebesgue measures on infinite-dimensional Frechet spaces. The question seems very natural in the context of creating a mathematically rigorous definition of the path integral of quantum field theory.
So: 1. Can an infinite-dimensional Frechet space have a measure which is locally finite, strictly positive and translation-invariant? 2. Can a separable infinite-dimensional Frechet space have a measure which is locally finite, non-zero and translation-invariant?
The motivation for the formulations of 1+2 are that the analogous statements for Banach spaces are false. 2 is more important for me since all interesting examples as separable, as far as I can tell.
If such measures exist, I'd be glad to get some references on whatever is known about them.