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 Banach spaces there is no a translation-invariant Borel measure in a separable Banach space 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 ) the then the answer to this question is yes.
