7
votes
2answers
483 views

Non-Hölder continuous devil's staircases

Let $f:[0,1]\to[0,1]$ be a devil's staircase in the usual sense. (That is, $f$ is continuous, non-decreasing, $f'=0$ on a set of full Lebesgue measure.) We also require the complement to the set where ...
21
votes
1answer
1k views

Analogues of Luzin's theorem

If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of ...
0
votes
3answers
205 views

how slow can the dimension of a product set grow?

Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$: $\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$, where $\sim$ denotes ...