Null Sets and Essentially Smooth Lipschitz Functions

Abstract

In this paper we extend the notion of a Lebesgue-null set to a notion which is valid in any completely metrizable Abelian topological group. We then use this definition to introduce and study the class of essentially smooth functions. These are, roughly speaking, those Lipschitz functions which are smooth (in each direction) almost everywhere.

Essentially Smooth Lipschitz Functions

“…We will say that a real-valued locally Lipschitz function f defined on a non-empty open subset A of X is essentially smooth on A if for each y # S(X ), [x # A : f 0 (x ; y){&f 0 (x ; &y)] is a Haar-null set. Using this definition the authors in [10,11] have extended some of the results in this paper to arbitrary Banach spaces.…”

Untitled

“…Christensen [11] defines the notion of 'Haar zero set' for abelian Polish groups, topological abelian groups with a complete separable metric. Borwein and Moors [10] generalize the work of Christensen by treating the nonseparable case.…”

Differentiability of cone-monotone functions on separable Banach space

“…We proceed to make the following definitions; for details on these, and other measure-related notions we use in the paper, we refer the reader to Benyamini and Lindenstrauss [1] (see also Borwein and Moors [3]). Indeed, while (a) of Theorem 1 is not especially well phrased for infinite dimensions, all the parts remain true, appropriately interpreted.…”

“…Let λ S be normalized Lebesgue measure on S. Then, consideration of the Borel measure defined by µ(C) = λ S (S ∩ C) shows that K is not µ-measurable and so is not universally measurable [3]. Proof.…”