Games in Banach Spaces: Questions and Several Answers

This is a preliminary sketch of several new ideas and questions concerning Aronszajn-null sets. 1. Motivation Aronszajn null sets were introduced by Aronszajn in the context of studying a.e. differentiability of Lipschitz mappings between Banach spaces. Christensen, Phelps and Mankiewicz studied the same problem independently and used Haar null, Gaussian null and cube null sets, respectively. See the monograph [1] for more information about the history. Csörnyei [2] proved that Aronszajn null, Gaussian null, and cube null sets coincide. It is well known that Haar null sets form a strictly larger family than Aronszajn null sets (see e.g. [1]). One of the questions in the differentiability theory is to understand the structure of the sets of points of Gâteaux nondifferentiability of Lipschitz mappings defined on separable Banach spaces. The strongest result in this context is due to Preiss and Zaj́ıček [3]. To that end, they introduced the Borel σ-ideal Ã. It follows from a recent result of Preiss that A = à in R, and it is unknown for 2 < dimX < ∞. In infinite dimensions, à ⊂ A and the inclusion is strict. It is also not known whether à = C̃ and C̃ ⊂ A (see [3] for the definitions). Understanding the structure of the sets of points of non-differentiability could possibly also be helpful in answering the longstanding open problem whether two separable Lipschitz isomorphic spaces are actually linearly isomorphic; this is known for some special Banach spaces, but is open for example for l1 and L1. We introduce a game-theoretic approach to Aronszajn null sets. It would be interesting to see whether this new perspective can yield interesting results which do not involve the new notions. For example, is it possible that this approach could help in answering the question in Problem 2.1? The second author is supported by the Koshland Fellowship. 1 2 JAKUB DUDA, BOAZ TSABAN 2. The Aronszajn-null game Let X be a separable Banach space (over R). The following definitions are classical: (1) For a nonzero x ∈ X, A(x) denotes the collection of all Borel sets A ⊆ X such that for each y ∈ X, A∩(Rx+y) has Lebesgue (one dimensional) measure zero. (2) A Borel set A ⊆ X is Aronszajn-null if for each dense sequence {xn}n∈N ⊆ X, there exist elements An ∈ A(xn), n ∈ N, such that A ⊆ ⋃ nAn. (3) A denotes the collection of Aronszajn-null sets. A is a Borel σ-ideal. Note that we are using a twisted version of the standard definition of Aronszajn-null sets, where {xn}n∈N is dense instead of just spanning. This is to make the treatment simpler. However, we do not know whether the two versions of the definition coincide. Problem 2.1 (folklore). Do we get an equivalent definition of Aronszajn-null sets if we replace “dense” by “complete” (i.e., having a dense span) in item (2)? The definition of Aronszajn-null sets motivates the following. Definition 2.2. The Aronszajn-null game G(A) for a Borel set A ⊆ X is a game between two players, I and II, who play an inning per each natural number. In the nth inning, I picks xn ∈ X, and II responds by picking An ∈ A(xn). This is illustrated in the following figure. I: x1 ∈ X x2 ∈ X . . . ց ր ց II: A1 ∈ A(x1) A2 ∈ A(x2) . . . I is required to play such that {xn}n∈N is dense in X. II wins the game if A ⊆ ⋃ n An; otherwise I wins. For a game G, the notation I↑ G is a shorthand for “I has a winning strategy in the game G”. Define I 6↑ G, II↑ G, II 6↑ G similarly. The following is easy to see. Lemma 2.3. If I6↑ G(A) for A, then A is Aronszajn-null. The converse is open. Conjecture 2.4. If A is Aronszajn-null, then I6↑ G(A) for A. Lemma 2.5. The property II↑ G(A) is preserved under taking Borel subsets and countable unions, i.e., it defines a Borel σ-ideal. GAMES IN BANACH SPACES: QUESTIONS AND SEVERAL ANSWERS 3 Proof. It is obvious that II↑ G(A) is preserved under taking Borel subsets. To see the remaining assertion, assume that B1, B2, . . . all satisfy II↑ G(A), and for each k let Fk be a winning strategy for II in the game G(A) played on Bk. Define a strategy F for II in the game G(A) played on ⋃ k Bk as follows. Assume that I played x1 ∈ X in the first inning. For each k let Ak,1 = Fk(x1), and set A1 = ⋃ k Ak,1 ∈ A(x1). II plays A1. In the nth inning we have (x1, A1, x2, A2, . . . , xn) given, where xn is the nth move of I. For each k let Ak,n = Fk(x1, Ak,1, x2, Ak,2, . . . , xn), and set An = ⋃ nAk,n ∈ A(xn). II plays An. Consider the play (x1, A1, x2, A2, . . . ). For each k, (x1, Ak,1, x2, Ak,2, . . . ) is a play according to the strategy Fk, and therefore Bk ⊆ ⋃ nAk,n. Consequently,