Let $f:[a,b]\to[0,\infty)$ ($\mathbb{R}\ni a<b\in\mathbb{R}$), and fix $c\geq0$. I want to establish the equivalence of the concepts of Lebesgue integrability and Henstock–Kurzweil integrability for this class of functions.
In particular, consider the following statements:
(1) For any $\varepsilon>0$, there exists a simple function $\omega=\sum_{j=1}^n z_j\mathbf{1}_{E_j}$ (where $n\in\mathbb{Z}_+$, the $\{z_j\}_{j=1}^n$ are distinct non-negative numbers, and the $\{E_j\}_{j=1}^n$ are Lebesgue measurable sets that form a partition of $[a,b]$) such that $0\leq\omega\leq f$ and
(1a) $f$ is Lebesgue measurable and $\sum_{j=1}^n z_j\mu(E_j)\in( c-\varepsilon,c]$, where $\mu$ is the Lebesgue measure, and the analogous sum of any non-negative simple function dominated by $f$ does not exceed $c$;
OR
(1b) $f$ is Lebesgue measurable and $\sum_{j=1}^n z_j\mu(E_j)> \varepsilon$.
(2) For any $\varepsilon>0$, there exists a “gauge” function $\delta_{\varepsilon}:[a,b]\to(0,\infty)$ such that for any $(x_j,t_j)_{j=1}^n\subset[a,b]$ ($n\in\mathbb{Z}_+$) satisfying
- $a<x_1<\ldots<x_{n-1}<x_n=b$,
- $t_j\in[x_j,x_{j-1}]$ for all $j\in\{1,\ldots,n\}$ (where $x_0\equiv a$), and
- $x_j-x_{j-1}<\delta_{\varepsilon}(t_j)$ for all $j\in\{1,\ldots,n\}$,
then
(2a) $\left|\sum_{j=1}^n f(t_j)(x_j-x_{j-1})-c\right|<\varepsilon$ for all $j\in\{1,\ldots,n\}$;
OR
(2b) $\sum_{j=1}^n f(t_j)(x_j-x_{j-1})>\varepsilon$ for all $j\in\{1,\ldots,n\}$.
I want to show that $(1a)\Longleftrightarrow(2a)$ and $(1b)\Longleftrightarrow(2b)$.
The tricky parts are as follows:
- For a given Lebesgue measurable function with integral $c$ (or $\infty$), how can one construct the desired gauge function so that the Henstock–Kurzweil integral is $c$ (or does not exist, respectively)?
- For a given Henstock–Kurzweil integrable function whose integral is finite or does not exist because it would be unbounded, how can one prove that it is Lebesgue measurable?
- For a given Henstock–Kurzweil integrable function, how can one construct the desired step function?
I skimmed through some of the relevant literature, but I failed to find any satisfying and not-too-abstruse explanation. (Disclaimer: I'm a newbie in HK-integration with some background in measure theory.) Thank you very much for your help in advance.
Note: (1a) basically states that $\int_{[a,b]}f(x)\,\mathrm{d}\mu(x)=c$ and (1b) that $\int_{[a,b]}f(x)\,\mathrm{d}\mu(x)=\infty.$
in abstract set up even an abolutely henstock kurzweil integrable function may not be measurable it all depends on whether1 is a measurable function. on euclidean spaces every henstock kurzweil integrable function is measurable. you can refer bartle. forvreal line.the proof depends on the second fundamnental theoe oof calculus which says that the primitive is differentiable almost everywhere. the theorem depends on vitalis lemma or austins lemma and bartles approach can be generalized to higher dimensions but we do not get primitive is differentiable but only symmetric derivative exists almost everywhere. please refer fremlins notes. however measurability can be deduced. a direct proof that positive lebesgue integrable function si henstock kurzweil integrable is there in lee peng yee rydolf vyborny integral aneasy approach . it relies on mct and dct. howebber hk integral is far more general . it does not require integrator to be even of bounded variation for stiltjes integral . measure theory is incapable of encompassing the generality An absolutely hk integrable Banach valued mapping on an interval [a, b] may not be measurable. though the evaluation theorem that is first fundamental theorem of calculus holds. Henstock has built of a theory of variation an anlaogue of outer measurre as an alternative to measure but the drux of the thory is not relying on notion of measurability and save fromm establishing measurability in applications. As Borel remarkedand criticized Lebesgue that he has bothere in intricate abstraction and dig the mountain to find a mole. today the basic role pof lebesgue thory is limited to proability theory . in practice we need close sets open sets which are all measurable