Henstock-Kurzweil versus Lebesgue: $f$ is Lebesgue integrable iff $f$ and $\mathrm{\mid }f\mathrm{\mid }$ are both HK integrable

Let $f:[a,b]\to \mathbb{R}$. Then $f$ is Lebesgue integrable on $[a,b]$ if and only if both $f$ and $\mathrm{\mid }f\mathrm{\mid }$ are Henstock-Kurzweil integrable on $[a,b]$, and in that case the two integrals of $f$ agree.

Equivalently: the Henstock-Kurzweil integral is a non-absolute integral, and $L^1[a,b]$ is exactly its absolutely integrable part. The inclusion is strict. The function $F(x)=x^2\sin (1\mathrm{/}{x}^2)$ for $x\mathrm{\ne }0$, $F(0)=0$, is differentiable everywhere on $[0,1]$ and $F^{\mathrm{\prime }}$ is HK integrable with ${\int }_0^1{F}^{\mathrm{\prime }}=F(1)-F(0)$, but $\mathrm{\mid }{F}^{\mathrm{\prime }}\mathrm{\mid }$ is not integrable in any sense, so $F^{\mathrm{\prime }}$ is not Lebesgue integrable. This is the point of the HK integral: it integrates every derivative, and the Newton-Leibniz formula ${\int }_a^b{F}^{\mathrm{\prime }}=F(b)-F(a)$ holds for every everywhere-differentiable $F$, with no hypothesis on $F^{\mathrm{\prime }}$ at all.

Not proved in this library. The comparison is recorded here; the HK integral itself is not deferred and is planned as ordinary content.

What would prove it. In one direction, a Lebesgue integrable $f$ is HK integrable with the same integral, and so is $\mathrm{\mid }f\mathrm{\mid }$, by the Vitali covering argument that produces gauges from measurable approximations. In the other, if $f$ and $\mathrm{\mid }f\mathrm{\mid }$ are both HK integrable then the indefinite HK integral of $\mathrm{\mid }f\mathrm{\mid }$ is absolutely continuous and monotone, and its derivative recovers $\mathrm{\mid }f\mathrm{\mid }$ almost everywhere, so $f\in L^1$. Both directions quantify over Lebesgue integrability (Lebesgue measure and the Lebesgue integral ‡) and use the differentiation theory (Lebesgue's differentiation theorem for monotone functions ‡, The sharp fundamental theorem of calculus (absolute continuity) ‡), which is why only the comparison is deferred.

Which page it serves. A Henstock-Kurzweil page in the integration track, which this library intends to build: the gauge integral needs only tagged partitions, a gauge $\delta :[a,b]\to (0,\mathrm{\infty })$, and Cousin's lemma, all of which are elementary and in scope. That page can prove the full Newton-Leibniz theorem for the HK integral, and then must record here what its relationship to $L^1$ is.

Why the comparison is the deferred part. The theorem is a statement about two integrals, one of which does not exist in this library. Stating it as a theorem would require the Lebesgue integral in the hypothesis and in the conclusion. The HK side loses nothing by the deferral: the improper integrals page and the fundamental theorems of calculus page can both use the gauge integral without mentioning measure at all.