Hake's theorem: a finite-endpoint generalized integral is a proper Henstock–Kurzweil integral after assigning the endpoint value

For the forward direction, [L3] restricts a proper HK integral to every compact prefix; apply [L2] to the one-cell partial partition $([c,b],b)$ after taking $c$ sufficiently close to $b$, and also make $\mathrm{\mid }f(b)\mathrm{\mid }(b-c)$ small, to obtain uniformly small tail integrals, so [L1] and [L3] make the truncation values converge to the proper integral.

For the reverse direction, let $A={\lim }_{c\uparrow b}{\int }_a^{c}f$ and set $c_i=b-(b-a)\mathrm{/}(i+1)$, so $c_0=a$ and $c_i\uparrow b$. By [L4], choose for each band $[c_{i-1},c_i]$ a gauge whose Saks–Henstock partial-partition error is below $\epsilon 2^{-i-4}$.

On each open band, take the minimum of its local gauge and half the distances to the two band endpoints; at each $c_i$, take the minimum of the adjacent gauges and half the adjacent band lengths. At $b$, choose a radius $\gamma$ so that the truncation tail error is below $\epsilon \mathrm{/}4$ and $\mathrm{\mid }f(b)\mathrm{\mid }\gamma <\epsilon \mathrm{/}4$. A partition fine for this global gauge can cross a band boundary only when tagged there, so it splits into finitely many complete band partitions, one final fine partial band partition, and a possible last cell tagged at $b$.

Additivity [L3], the summable local error budget from step 1.2, the Saks–Henstock estimate on the final partial band, the truncation bound, and the endpoint-cell bound from step 2.1 show that every fine sum differs from $A$ by less than $\epsilon$. Thus the extension is properly HK integrable with integral $A$. Changing the assigned value at $b$ alters only the last endpoint-tagged term, whose length the gauge can make arbitrarily small, so the integral is independent of that value; reflection gives the left-endpoint form.