Bruckner--Garg-type results with respect to Haar null sets in $C[0,1]$

Abstract

A set $\mathcal{A}\subset C[0,1]$ is \emph{shy} or \emph{Haar null } (in the sense of Christensen) if there exists a Borel set $\mathcal{B}\subset C[0,1]$ and a Borel probability measure $\mu$ on $C[0,1]$ such that $\mathcal{A}\subset \mathcal{B}$ and $\mu\left(\mathcal{B}+f\right) = 0$ for all $f \in C[0,1]$. The complement of a shy set is called a \emph{prevalent} set. We say that a set is \emph{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/non-shy many $f\in C[0,1]$? The classical Bruckner--Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) $f\in C[0,1]$ from the topological point of view. We prove that the functions $f\in C[0,1]$ for which the same characterization holds form a Haar ambivalent set. In an earlier paper we proved that the functions $f\in C[0,1]$ for which positively many level sets with respect to the Lebesgue measure $\lambda$ 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 $f\in C[0,1]$ for which positively many level sets with respect to the occupation measure $\lambda\circ f^{-1}$ are not perfect form a Haar ambivalent set in $C[0,1]$. We show that for the prevalent $f\in C[0,1]$ for the generic $y\in f([0,1])$ the level set $f^{-1}(y)$ is perfect. Finally, we answer a question of Darji and White by showing that the set of functions $f \in C[0,1]$ for which there exists a perfect $P_f\subset [0,1]$ such that $f'(x) = \infty$ for all $x \in P_f$ is Haar ambivalent.