Aronszajn null and Gaussian null sets coincide
In the study of Gateaux differentiability of Lipschitz maps from separable Banach spaces into various types of spaces several authors have introduced notions of smallness for sets and have proved such functions are differentiable outside of an exceptional small set. The results of this type referenced in the present paper are: \textit{N. Aronszajn}, Stud. Math. 57, 147-190 (1976; Zbl 0342.46034); \textit{J. P. R. Christensen}, Publ. Dépt. Math., Lyon 10, No. 2, 29-39 (1973; Zbl 0302.43001); \textit{P. Mankiewicz}, Stud. Math. 45, 15-29 (1973; Zbl 0246.46002) and \textit{R. R. Phelps}, Pac. J. Math. 77, 523-531 (1978; Zbl 0396.46041)]. Three of these notions are: Aronszajn null sets; nondegenerate cube measure null sets; and Gaussian null sets. These are \(\sigma\)-ideals of sets \(\Sigma_1\), \(\Sigma_2\), and \(\Sigma_3\) respectively definable in any separable Banach space. It was previously known that \(\Sigma_1\subset (\Sigma_2 \cap \Sigma_3)\). The present author shows that \(\Sigma_1=\Sigma_2=\Sigma_3\).
scientific article; zbMATH DE number 1356476