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

Richárd Balka, Udayan B. Darji+1 more|Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)|Nov 21, 2013

Advanced Topology and Set Theory6 references9 citations

TL;DR

This paper investigates the topological structure of level sets of continuous functions on [0,1] under the prevalence (Haar null) measure-theoretic framework. It establishes that the set of functions satisfying the Bruckner–Garg Theorem—whose level sets are countable unions of Cantor sets, singletons, or their combinations—is neither shy nor prevalent, i.e., Haar ambivalent, revealing a fundamental difference between Baire category and prevalence in describing typical continuous functions.

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 $μ$ on $C[0,1]$ such that $\mathcal{A}\subset \mathcal{B}$ and $μ\left(\mathcal{B}+f ight) = 0$ 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 $λ$ 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 $f\in C[0,1]$ for which positively many level sets with respect to the occupation measure $λ\circ f^{-1}$ 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.

Motivation & Objective

• To determine the prevalence status of functions in C[0,1] whose level sets satisfy the Bruckner–Garg Theorem’s topological characterization.
• To compare topological properties of level sets under Baire category (generic) versus prevalence (typical in measure-theoretic sense).
• To investigate whether sets of functions with certain level set structures—such as those with non-perfect fibers or infinite derivatives on perfect sets—are shy, prevalent, or Haar ambivalent.
• To resolve a question by Darji and White on the prevalence of functions with infinite derivative on a perfect set in [0,1].

Proposed method

• Uses the concept of Haar null (shy) and prevalent sets in the Polish group C[0,1] under the supremum norm, following Christensen’s definition.
• Applies measure-theoretic tools: constructs Borel probability measures μ and Borel sets B such that μ(B + f) = 0 for all f ∈ C[0,1], to define shy sets.
• Employs occupation measures λ∘f⁻¹ to analyze level sets with respect to the distribution of f-values, rather than Lebesgue measure.
• Applies results from random processes (e.g., Brownian motion) and functional analysis to construct functions with specific level set behavior.
• Uses a recursive construction of nested intervals In,σ with lengths ln to define a continuous function g such that g′(x) = ∞ on a perfect set P.
• Applies Dougherty’s result on fibers of prevalent maps from Cantor sets to R^d to infer that generic fibers have cardinality continuum.

Experimental results

Research questions

• RQ1 Is the set of functions satisfying the Bruckner–Garg Theorem (in terms of level set structure) prevalent, shy, or Haar ambivalent in C[0,1]?
• RQ2 What is the prevalence status of functions for which positively many level sets (with respect to Lebesgue measure) are singletons?
• RQ3 What is the prevalence status of functions for which positively many level sets (with respect to the occupation measure λ∘f⁻¹) are not perfect?
• RQ4 Is the set of functions with infinite derivative on a perfect set in [0,1] prevalent, shy, or Haar ambivalent?
• RQ5 How does the fiber structure of prevalent functions f ∈ C(K, R^d) compare to the generic case, especially for compact metric spaces K without isolated points?

Key findings

• The set of functions f ∈ C[0,1] satisfying the Bruckner–Garg Theorem is Haar ambivalent, meaning it is neither shy nor prevalent.
• The set of functions f ∈ C[0,1] for which there exist positively many y ∈ R such that f⁻¹(y) is a singleton (with respect to Lebesgue measure) is Haar ambivalent.
• The set of functions f ∈ C[0,1] for which there exist positively many y ∈ R such that f⁻¹(y) is not perfect (with respect to the occupation measure λ∘f⁻¹) is Haar ambivalent.
• For the prevalent f ∈ C[0,1], the generic y ∈ f([0,1]) has a level set f⁻¹(y) that is perfect.
• 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.
• For the prevalent f ∈ C(K, R^d) with K a compact metric space without isolated points, there exists a non-empty open set U_f ⊂ R^d such that f⁻¹(y) has cardinality 2^ℵ₀ for all y ∈ U_f.