Royden's classical Vitali-covering proof is a third route and is not run on this page

There is a third classical proof that monotone functions are differentiable almost everywhere. One first proves measurability of the Dini derivative level sets, then applies the Vitali covering theorem to the sets where the upper and lower derived numbers disagree.

This page does not run that route. The rising-sun proof and the Lebesgue-Stieltjes proof already expose the mathematical comparison the design wanted:

• the rising-sun route reaches the theorem without the Vitali covering theorem;
• the measure route factors the result through differentiation of measures and the Stieltjes correspondence;
• the Vitali route is historically important, but it duplicates the almost-everywhere differentiability conclusion without adding a later dependency that this page needs.