Two-interval approximation by one interval (${\inf }_{h\text{ interval}}{\mathrm{err}}_D(h)\le \frac{1}{3}$)

For a target positive set consisting of two disjoint intervals of total probability $p$, choosing its heavier component has misclassification risk at most $p/2$. Choosing the spanning interval has misclassification risk at most $1-p$. The smaller bound is at most $1/3$. The constant is sharp: on three ordered atoms with equal masses and labels $1,0,1$, every single interval makes a mistake on at least one atom. Together with the VC uniform convergence theorem, this gives weak PAC learning using an interval hypothesis class.