How large is the small if the space is big? (Null sets in Banach spaces) (Q2714396)

Contrary to the finite dimensional setting, there is no distinguished translation invariant measure in infinite dimensional Banach spaces. Therefore, it is of fundamental importance to clarify how various measures relate to each other.NEWLINENEWLINEThis paper surveys and summarizes the results on how the important concepts of negligible sets can be compared in the Banach space setting. The notions of Haar, Gauss, cubic, and Aronszajn null sets are recalled and also their basic properties are investigated in the first part of the paper.NEWLINENEWLINEIt was proved by Phelps in 1978 that Haar null sets are automatically Gauss null sets but the converse is not true. Analogously, Aronszajn null sets are negligible in the Gaussian sense. It is immediate to see that Aronszajn null sets are cubic null sets as well. The main goal of this paper is to solve the long standing open problem if the latter two implications are reversible. More precisely, it is shown that in separable Banach spaces the notions of Gaussian, cubic, and Aronszajn null sets coincide. These new results are also published in another article of the author [Isr. J. Math. 111, 191--201 (1999; Zbl 0952.46030)].