**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.



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> **Question:** Is the set of zeroes of a Brownian motion almost surely a mesoscopic set?
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*Remark: Note that there exist mesoscopic sets of Lebesgue measure zero - for example the Cantor set with $f$ being the Cantor staircase function.*



Largeness of the set of zeroes of a Brownian motion
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>


