Let $f\,$ be a bounded function and $D=\bigcup D_\alpha$ be a measure zero set where
$D_\alpha :=\{x_0\in[a,b]: \omega_f(x_0)\geq \alpha, \},\quad \alpha >0. $ Setting $\alpha := \frac{\epsilon}{2(b-a)}$, it needs to be shown that there exists a finite collection of open intervals $\{G_1,G_2,...,G_N\}$ such that the collection covers $D_\alpha$ and has $\sum_{n=1}^N |G_i|\leq \frac{\epsilon}{4M}$ where $|G_i|$ is the length of the interval $G_i$ and $M = \sup\{f(x): x\in [a,b]\}$.