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 ?

The negative result mentioned in the comment by Zen Harper, is about invariant measures, nonexistent on infinitedimensional 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/94shiftedKuelbsLiLinde.pdf 


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. 


We can give a construction of a standard translationinvariant Borel measure in $R^N$(here $N$ denotes a set of all naturall numbers), which obtains the value one on the infinitedimensional cube $[0;1[^N$. Actually, we are free from the demand of sigmafiniteness, because the space $R^N$ is covered by the uncountable family of pairwise disjoint shifts of $[0;1[^N$. Measures with abovementioned properties are adopted as partial analogs of the Lebesgue measure in the infinitedimensional topological vector space $R^N$. Partial analogs of the Lebesgue measure in general Banach spaces are assumed as translationinvariant Borel measures which obtain the numerical value one on the unit sphere or on the standard infinitedimensional 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 nonseparable Banach spaces. I have considered the following problem posed by C. Rogers (1998): Does there exist a such translationinvariant Borel measure in $\ell^{\infty}$ which obtains the numerical value one on the closed unite sphere?( here $\ell^{\infty}$ denotes a nonseparable Banach space of all bounded realvalued 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 σfinite measures and some of their applications Acta Mathematica Sinica, English Series (2011) 27: 477496, March 01, 2011" You have mentioned that in separable Banach spaces there is no a translationinvariant Borel measure which obtain a numerical value one on the unite ball. But if we consider a question asking whether there is a translationinvariant Borel measure in a separable Banach space which obtain a numerical value one on the infinitedimensional parallelepiped ( generated by any Markushewicz basis, in particular, by Schauder basis ) then the answer to this question is yes. 

