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

Abstract

AbstractA setisshyorHaar null(in the sense of Christensen) if there exists a Borel setand a Borel probability measureμonC[0, 1] such thatandfor allf∈C[0, 1]. The complement of a shy set is called aprevalentset. We say that a set isHaar ambivalentif 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/non-shy manyf∈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 functionsf∈C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper, Balkaet al. proved that the functionsf∈C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set inC[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functionsf∈C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦f–1are not perfect form a Haar ambivalent set inC[0, 1].We show that for the prevalentf∈C[0, 1] for the genericy∈f([0, 1]) the level setf–1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functionsf∈C[0, 1] for which there exists a perfect setPf⊂ [0, 1] such thatfʹ(x) = ∞ for allx∈Pfis Haar ambivalent.