Equivalence of the Lebesgue integral and the Henstock–Kurzweil integral on nonnegative real functions

Due to the completely different nature of the Riemann and Lebesgue approaches, there is no direct proof of the equivalence you mention as far as I know.

To show that the class of absolutely HK-integrable functions is the class of Lebesgue integrable functions, you use the following characterisation:

If $V$ is a vector space of Lebesgue measurable functions, and if $I : V \to \mathbb{R}$ is a linear map satisfying the following properties:

• $C([a, b]) \subset V$;
• $f \in V$ implies $|f| \in V$;
• if $(f_n)_n \subset V$ satisfy the assumptions of the monotone convergence theorem, then $\lim_n f_n \in V$ and $I(\lim_n f_n) = \lim_n I(f_n)$

then $V = L^1([a,b])$ and $I$ is the Lebesgue integral.

Therefore, you need to show:

• Riemann integrable functions are HK integrable;
• HK satisfies the Monotone convergence theorem;
• HK integrable functions are measurable.

The second one is classical and can be found in any book on the HK integral. The measurabilit part can be shown by approximating the function by step functions given by the Riemann sums. The proof is a bit technical though (but not too much).

I'll give you an outline of how to show that not only a function that is Lebesgue integrable is also Henstock Integrable and the integrals have the same value, but also that the same thing holds for a Lebesgue-Stieltjes and Henstock-Stieltjes integrals. You can find a very well written definition of the Henstock-Stieltjes integral in the Handbook of Analysis and Its Foundations by Eric Schechter[1]. This proof is far more difficult than a simple calculation.

You'll first need to prove some technical lemmas:

1) $$\int_a^b f d\varphi=\int_a^c f d\varphi+\int_c^b f d\varphi$$ for $a<c<b$. This is not so easy to prove as it's easily proved for Riemann or Lebesgue integrals. Hint: use the gauge $\gamma(t)=1$ if $t=c$ and, otherwise, $\gamma(t)=\min\{1,|1-t|\}$.

2) now you proceed to prove the Henstock-Saks lemma using (1), which is a very important result and it's used to prove a lot of further theorems about Henstock (and Henstock-Stieltjes) integrals.

3) and for the most difficult part you now can prove the monotone convergence theorem, this proof is very far from trivial, you should check it at A Modern Theory of Integration by Bartle[2].

4) you also have to prove some minor results such that the Henstock-Stieltjes integration is linear (that's not difficult) and that if a subset of a closed interval $S$ is a countable union of intervals, then its characteristic function is Henstock-Stieltjes integrable.

5) this is the last technical lemma: let $f$ and $\varphi$ be functions with domain $[a,b]$, the first is nonnegative and the second is increasing and the Henstock-Stieltjes integral $\int_a^b fd\varphi$ exists. Let $\varepsilon>0$ be given and $E=\{t\in [a,b]| f(t)\geq 1\}$, then exists an open set $G$ such that $E\subset G$ and $$\int_a^b \chi_G d\varphi \leq \varepsilon +\int_a^b fd\varphi.$$ You can find a proof for this lemma in [1], it's not an easy one.

Now you can prove the main result (with the lemmas I stated you can check the proof at [1] and don't be lost):

Let $\varphi$ be a increasing function and $\mathcal{K}$ a collection of all sets $S\subset [a,b]$ such that the Henstock-Stieltjes of $\chi_S$ in $[a,b]$ exists, then $\mu_\varphi (S)=\int_a^b \chi_S d\varphi$ where $\mu_\varphi(S)$ is the Lebesgue-Stieltjes measure of the set $S$. Furthermore $\mathcal{K}$ is a complete $\sigma$-algebra containing the Borel sets $\mathcal{B}$ of $[a,b]$ and $([a,b],\mathcal{K},\mu_\varphi)$ is a complete measure space that is the completation of $([a,b],\mathcal{B},\mu_\varphi)$. Even more: every positive finite measure $\mu$ over $\mathcal{B}$ is the same as $\mu_\varphi$ for some increasing function $\varphi$.

With that you can show that any Lebesgue-Stieltjes integrable function is also Henstock-Stieltjes integrable end that their integrals have the same value, to prove that you will use the monotone convergence theorem plus the fact they are the same for the characteristic function. At last, you also get the converse and finaly the result you were waiting for: that any positive function that is Henstock-Stieltjes integrable is also Lebesgue-Stieltjes and their integral is equal by the same argument. A good remark is that this won't be true to all functions, in [1] and [2] we can find counter-examples to that.