Source clarifications: approximately fair allocations

Source: On Approximately Fair Allocations of Indivisible Goods, Theorem 2.3 and Lemma 2.4.

Lemma 2.4: choosing a progressing endpoint

• Lemma 2.4’s minimum endpoint at which every player values [0,x] by at most alpha → a finite high-point/residual partition. The minimum can be x=0, so the printed step need not advance. The replacement constructs pieces worth at most alpha to every player and applies the finite allocation theorem.
• For finitely many finite valuation measures mu_i on a common bounded interval, with every atom at most alpha>0, the result is an allocation A satisfying mu_i(A_j)-mu_i(A_i)<=alpha for all players i,j. The construction does not prove Lemma 2.4’s O(n/alpha) piece-count bound, where n is the player count. Theorem 2.3’s alpha=0 branch still depends on the cited external envy-free allocation theorem.