While thinking about what some measure on an infinite dimensional Banach space could look like a came across the point that if I'd like to assign a size to all epsilon balls, they by Riesz' lemma necessarily will have to have either zero or infinite size. But then every bounded set will have zero or infinite size too. Moreover every countable union of these will have zero or infinite size as well...
Now I'm wondering what nontrivial finite(!) translation invariant Borel measures on Banach spaces there are. Do you have some concrete examples?
Putting measures on infinite-dimensional spaces is hard and an interesting problem of current research. There is, for the same reasons you point out, no translation-invariant measure -- no analogue of Lebesgue measure -- on an infinite dimensional space. Probability is a good source of examples for such measures: Wiener measure is the prototypical example of a non-trivial finite, Borel-regular measure on the Banach space of continuous functions with the supremum norm.