A set
is shy or Haar null (in the sense of Christensen) if there exists a Borel set
and a Borel probability measure ΞΌ on C[0, 1] such that
and
for all f β 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/non-shy many f β C[0, 1]?
The classical BrucknerβGarg theorem characterizes the level sets of the generic (in the sense of Baire category) f β C[0, 1] from the topological point of view. We prove that the functions f β C[0, 1] for which the same characterization holds form a Haar ambivalent set.
In an earlier paper, Balka et al. proved that the functions f β C[0, 1] for which positively many level sets with respect to the Lebesgue measure Ξ» 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 β C[0, 1] for which positively many level sets with respect to the occupation measure Ξ» β¦ fβ1 are not perfect form a Haar ambivalent set in C[0, 1].
We show that for the prevalent f β C[0, 1] for the generic y β f([0, 1]) the level set fβ1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functions f β C[0, 1] for which there exists a perfect set Pf β [0, 1] such that fΚΉ(x) = β for all x β Pf is Haar ambivalent.