Bruckner–Garg-Type Results with Respect to Haar Null Sets inC[0, 1]

Abstract

Abstract. A set A ⊂ C[0, 1] is shy or Haar null (in the sense of Christensen) if there exists a Borel set B ⊂ C[0, 1] and a Borel probability measure µ on C[0, 1] such that A ⊂ B and µ (B + f ) = 0 for all f ∈ C[0, 1]. The complement of a shy set is called a prevalent set. We say that a set is Haar ambivalent if it is neither shy nor prevalent.The main goal of the paper is to answer the following question: What can we say about the topological properties of the level sets of the prevalent/nonshy many f ∈ C[0, 1]?The classical Bruckner-Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) f ∈ C[0, 1] from the topological point of view. We prove that the functions f ∈ C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper we proved that the functions f ∈ C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set in C[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions f ∈ C[0, 1] for which positively many level sets with respect to the occupation measure λ • f −1 are not perfect form a Haar ambivalent set in C[0, 1].We show that for the prevalent f ∈ C[0, 1] for the generic y ∈ f ([0, 1]) the level set f −1 (y) is perfect.Finally, we answer a question of Darji and White by showing that the set of functions f ∈ C[0, 1] for which there exists a perfect set P f ⊂ [0, 1] such that f ′ (x) = ∞ for all x ∈ P f is Haar ambivalent.

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

“…For the following well-known lemmas see e.g. [44,Lemma 4] and [5], respectively. Note that Lemma 4.20 is stated in [5] only in the case K = [0, 1], but the proof works verbatim for all K. There is a compact set K ⊂ R such that dim H K = dim P K = 1 and…”

“…For the prevalent f ∈ C[0, 1] the sets f −1 (min f ) and f −1 (max f ) are singletons, see e.g. [5]. The aim of this section is to prove that all other non-empty level sets can be of Hausdorff dimension one.…”

Haar null and Haar meager sets: a survey and new results

“…Let us also mention a few results that are measure theoretic duals of classical results of real analysis about generic continuous functions: [60, 61, 89] concern nowhere differentiable functions, and [5] is the Bruckner–Garg theorem describing the topological structure of the level sets. Interestingly, in the case of [5] and [89] the generic behavior happens with ‘probability strictly between 0 and 1’, that is, the set of functions exhibiting the behavior in question is neither Haar null nor co‐Haar null.…”

Haar null sets without G δ hulls

“…For more results concerning fundamental properties and applications of Haar null sets in non-locally compact groups see e.g. [1], [2], [10], [11], [12], [14], [16], [19], [23], [24].…”