Measures on infinite dimensional Banach spaces Does there exist a Borel measure or any valid measure on an infinite dimensional Banach space such that a bounded open set in this space has a positive measure ?
 A: It is a consequence of Riesz' Lemma that every open ball in an infinite dimensional normed space contains a disjoint sequence of smaller open balls. They all have the same measure under a translation invariant measure, so if the surrounding ball has finite measure, they all have measure zero. For separable spaces, this shows that every open set gets either measure 0 or $\infty$ under a translation invariant measure.
If you don't care about translation invariance, Wiener measure on the space of continuous function on [0,1] with starting value 0 should satisfy your condition.
A: We can give a construction of a standard translation-invariant Borel measure in
$R^N$(here $N$ denotes a set of all naturall numbers), which obtains the value one on the infinite-dimensional cube $[0;1[^N$. Actually, we are
free from the demand of sigma-finiteness, because the space $R^N$ is covered by the uncountable
family of pairwise disjoint shifts of $[0;1[^N$. Measures with above-mentioned properties are adopted as partial analogs of the Lebesgue measure in the infinite-dimensional topological vector space $R^N$.  Partial analogs of the Lebesgue measure in general Banach spaces are assumed as translation-invariant Borel measures which obtain the numerical value one on the unit sphere or on the standard infinite-dimensional parallelepiped ( generated by any basis ).
The fundamental works of English mathematicians C. Rogers  and D.
Fremlin  are devoted to problems of the existence of such measures in non-separable Banach spaces. I have considered
the following problem posed by C. Rogers (1998):
Does there exist a such translation-invariant Borel measure in $\ell^{\infty}$ which obtains the
numerical value one on the closed unite sphere?( here $\ell^{\infty}$ denotes a non-separable Banach space of all bounded real-valued sequences equipted with standard norm)
My result asserts that this question is not solvable within the the theory $ZF+DC$.
On the one hand, we can construct a "consistent" extension of the theory  $ZF+DC$ where this question is solvable positivelly( such a theory is the so called "Solovay model")
On the other hand, we can construct a "consistent" extension of the theory  $ZF+DC$ where this question is solvable negativelly ( such a theory is "ZF+AC+"there is no a measurable cardinal")
The proof of these facts can be found in 
"G.R.Pantsulaia, On ordinary and standard products of infinite family of σ-finite measures and some of their applications 
Acta Mathematica Sinica, English Series (2011) 27: 477-496, March 01, 2011"
You have mentioned that in separable Banach spaces there is no a translation-invariant Borel measure  which obtain a numerical value one on the unite ball. But if we consider a question asking whether there is a translation-invariant Borel measure in a separable Banach space which obtain a numerical value one on the infinite-dimensional parallelepiped ( generated by any Markushewicz  basis, in particular, by Schauder basis ) then the answer to this question is yes.
A: The negative result mentioned in the comment by Zen Harper, is about invariant measures, non-existent on infinite-dimensional Banach spaces. If one does not require the invariance, there is no problem. See, for example, the calculation of the Gaussian measure of a ball in the paper
http://titan.math.udel.edu/~wli/papers/94-shifted-Kuelbs-Li-Linde.pdf
