关于$C[0,1]$中Haar空集的Bruckner-Gark型结果
Bruckner--Garg-type results with respect to Haar null sets in $C[0,1]$
摘要：
一个集合$\mathcal{A}\subset C[0,1]$是\emph{shy}或\emph{Haar null }（在克里斯滕森的意义下），如果存在一个Borel集合$\mathcal{B}\subset C[0,1]$和一个Borel概率测度$μ$在$C[0,1]$上，使得$\mathcal{A}\subset \mathcal{B}$和$μ\left(\mathcal{B}+f\right) = 0$对于所有的$f \in C[0,1]$。害羞集的补集称为\emph{prevalent}集。我们说一个集合是\emph{Haar ambivalent}，如果它既不害羞也不普遍。该文件的主要目标是回答以下问题：我们可以说什么流行的/非害羞的许多$f\in C[0,1]$的水平集的拓扑性质？经典的Bruckner-Garg定理从拓扑的角度刻画了一般（Baire范畴意义下）$f\in C[0,1]$的水平集.我们证明了$f\in C[0,1]$的功能，相同的表征持有形成一个哈尔矛盾集。在较早的文件中，我们证明了功能$f\in C[0,1]$积极的许多水平集关于勒贝格措施$λ$是单例形式的非害羞的$C[0,1]$。上述结果表明，这个集合实际上是Haar矛盾集。现在我们证明了函数$f\in C[0,1]$的积极许多水平集关于占领措施$λ\circ f^{-1}$是不完善的形式Haar矛盾集$C[0,1]$。我们发现，普遍的$f\in C[0,1]$的通用$y\in f([0,1])$的水平集$f^{-1}(y)$是完美的。最后，我们回答了一个问题，Darji和白色的显示，一组功能$f \in C[0,1]$，存在一个完美的$P_f\subset [0,1]$，使$f'(x) = \infty$的所有$x \in P_f$是哈尔矛盾。
英文摘要：
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 $μ\left(\mathcal{B}+f\right) = 0$ 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 $λ$ 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.
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