TITLE: Largeness of the set of zeroes of a Brownian motion
QUESTION [5 upvotes]: Definitions:
A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero Lebesgue measure.

Question: Is the set of zeroes of a Brownian motion almost surely a mesoscopic set?

Remark: Note that there exist mesoscopic sets of Lebesgue measure zero - for example the Cantor set with $f$ being the Cantor staircase function.

REPLY [9 votes]: Yes, the local time (at zero) maps the zero set of Brownian motion to an interval. See e.g. Lemma 6.9 page 159 in [1] for continuity.
[1] Brownian motion, by Peter Mörters and Yuval Peres. Cambridge University Press, 2010   https://people.bath.ac.uk/maspm/book.pdf