The Saks–Henstock lemma for fine partial tagged partitions

The empty family has error $0$. Otherwise fix a whole-interval gauge whose full-partition error is below $\epsilon \mathrm{/}4$. After a partial partition fine for that fixed gauge is given, [L2] makes $f$ integrable on each of its finitely many complementary compact intervals. For any prescribed complement error, [L3] supplies a local accuracy gauge there; [L1] supplies a partition fine for the minimum of that local gauge and the already fixed whole-interval gauge. Thus the resulting completions are both arbitrarily accurate and fine for the original gauge.

For the cells with nonnegative local error, complete their complement with fine partitions whose total local error is below $\epsilon \mathrm{/}4$. Additivity [L2] identifies the resulting full-partition error with the selected positive errors plus those complement errors, so the positive total is below $\epsilon \mathrm{/}2$. Repeating the construction for the negative cells bounds the absolute value of their total by $\epsilon \mathrm{/}2$; adding the two bounds gives the displayed strict estimate.