**Question.** There are several ways of defining a notion of a ‘zero measure’ subset of a Banach space $X$. Which one is the ‘right’ or failing that, the preferred notion? (See below for a more precise reformulation of the question.)

Let $X$ be an infinite-dimensional Banach space. The obvious problem when trying to extend the notion of negligible (or ‘zero measure’) subsets from the finite-dimensional setting is that often $X$ will not admit a natural measure. (In contrast $mathbf{R}^n$ has the Lebesgue measure.)

There are several approaches that circumnavigate this issue, for example:

*Haar null sets*are those subsets $A subset X$ for which there is a Borel probability measure $mu$ on $X$ so that every translate of $A$ has $mu$-measure zero: $mu(A – x) = 0$ for all $x in X$, which apparently go back to Christensen (1).- There are
*$Gamma$-null sets*as defined by Lindenstrauss and Preiss,*Gaussian null sets*,*Aronszajn null sets*and more. A discussion of the various notions is given in a paper of Bogachev (2).

I am interested in results for function spaces—say $X = C^{k,alpha}$ is a Holder space for example—and where one tries to prove that some property holds for ‘almost every’ map.

**Question’.** Is one of these notions preferred over the others in this context? Is there a rule of thumb when to use which one?

(1) J.P.R. Christensen. On sets of Haar measure zero in abelian Polish groups, *Israel J. Math.* **13** (1972), 255-260.

(2) V.I. Bogachev. Negligible sets in infinite-dimensional Banach spaces. *Analysis Math.*, **44** (3) (2018), 299–323.