A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a *[Brjuno number](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Continued_fraction#Infinite_continued_fractions_and_convergents) $\frac{p_i}{q_i}$ of the [continued fraction](https://en.wikipedia.org/wiki/Continued_fraction) expansion of $\alpha$.

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