The Fourier–Faber–Schauder series unconditionally divergent in measure

We prove that, for every $\varepsilon\in (0,1)$, there is a measurable set $E\subset[0,1]$ whose measure $|E|$ satisfies the estimate $|E|>1-\varepsilon$ and, for every function $f\in C_{[0,1]}$, there is $\tilde f\in C_{[0,1]}$ coinciding with $f$ on $E$ whose expansion n the Faber–Schauder system diverges in measure after a rearrangement.