# fa.functional analysis – What is the ‘right’ definition of zero measure subsets of Banach spaces?

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.