A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator of the $i$th convergent $\frac{p_i}{q_i}$ of the continued fraction expansion of $\alpha$.

Is the set $B$ of Brjuno numbers measurable, and if yes, what is the measure of $B\cap[0,1]$?