Are there any unsolved Problems in Theory of Integration? Ever since the development of Theory of integration. each specific problem has led to development of new integrals.


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*The notion of integrability of continuous functions (Cauchy integral) led to Riemann Integral, where the function is still integrable if it has finite number of discontinuities.

*Failure of convergence theorem and a weaker FTC led to Lebesgue integral.

*Improper integral and certain non-Lebegsue Integrable functions led to the development of HK(guage) Integrals. 
Question 1. Have these  developments exhausted integration theory?
Question 2. Or there are still some shortcomings or unsolved problems in this area?
 A: I want to give an example that came up within my master thesis: integration of internal functions over convex domains in non-standard analysis.
The set of  hyperreals ${^*\mathbb R}$ is an ordered field extending the reals which contains infinitesimal ($|x|<\tfrac 1n$ for all $n\in\mathbb N$) and unlimted numbers ($|x|>n$ for all $n\in\mathbb N$). An internal function ${^*f:{}^*\mathbb R \to {}^*\mathbb R}$ arises as a extension of a sequence $f_n$ of ordinary function to unlimited numbers. As an example 
$$f_\epsilon(x) = \frac{1}{\sqrt{2\pi}\epsilon}\exp(-\frac{1}{2}\frac{x^2}{\epsilon^2})$$
can be extended to infinitesimal $\epsilon>0$ whereupon it acts like a the dirac delta, in the sense that for any compactly supported, smooth test function $\varphi$, 
$$ \int f_\epsilon(x) \varphi(x) dx \simeq \varphi(0)$$
where $\simeq$ means that the quantities only differ by an infinitesimal amount. This type of integral is easier to rigorize because we integrate over a compact domain. However one would like to integrate over more general convex domains in $^*\mathbb R$.

Problem: Develop a theory that allows the evaluation of integrals of the form $\int_C {^*f(x)} dx$, where $^*f$ is internal and $C\subset{^*\mathbb R}$ is convex.

There are some obvious problems. For example what should $\int_{\mathbb I} 1 dx$ be? Here $\mathbb I$ is the set of infinitesimals. There is no largest infinitesimal, yet at the same time the integral is 'obviously' bounded by any positive non-infinitesimal number. 
A way out of this dilemma is, instead of letting the integral be a map onto $^*\mathbb R$, to consider a sort of fuzzy integral that maps a function onto an external number $\alpha \in \mathbb E$.  An external number is nothing but a pair $(\alpha_0,\mathcal N)$, where $\alpha_0\in{^*\mathbb R}$ is the value of the integral and $\mathcal N\subset {}^*\mathbb R$ is sort of it's uncertainty. In the above case we'd have $\alpha_0 = 0$ and $\mathcal N = \mathbb I$.
Here the set $\mathcal N$ is always a so called neutrix, that is a convex additive subgroup of ${^*\mathbb R}$. These sets can have very peculiar properties. This theory of integration is called external integration and was developed by I. van den Berg and Fouad Koudjeti.
Neutrices, external numbers, and external calculus (unfortunately paywalled)
Some more background on external numbers can be found here
One of the the issues with this theory is however that it is rather complicated and only can handle non-negative functions. I  believe that it should be possible to develop a 'neat' nonstandard integration theory that addresses these issues.
Finally I want to  give a nice example of how one can intuitively work with nonstandard integration which is also from the cited book but translated into a more approachable language. 
We use a nonstandard version of Laplace's method used to prove Stirling's formula $n!\sim \sqrt{2\pi n} (\frac ne)^n$: By the gamma function representation, $n! = \int_0^\infty t^n e^{-t}dx$. Instead of $n$, we insert an unlimited hypernatural number $N$. Note that the integrand has a unique global maximum at $t=N$. Doing a coordinate transformation $x=t/N-1$ we obtain
$$N! =  \int_{-1}^\infty e^{-N(x+1)}N^{N+1} (x+1)^N dx = N^{N+1}e^{-N}
\int_{-1}^\infty e^{-N(x-\log(1+x))}dx $$
Here note that the integrand is infinitesimal whenever $x-\log(1+x)$ is not infinitesimal itself. In fact the only non-negligible contribution can occur when $-N(x-\log(1+x)) \sim -\tfrac 12 Nx^2 +O(x^3)$ is not infinitesimal, which is the case when $x\in \frac{1}{\sqrt N} \mathbb L$, where $\mathbb L$ is the set of limited numbers. ($|x|<n$ for some $n\in\mathbb N$). Hence we can restrict the integral to this set. By doing so we make a slight error in the form of $N! = (1+\epsilon)\cdot\ldots$ with $\epsilon$ infinitesimal. 
In fact an equivalence relation $\asymp$ is given on the hyperreals by $x\asymp y \iff x \in y(1+\epsilon)$ for some infinitesimal $\epsilon$.
$$ N! \asymp N^{N+1} e^{-N} \int_{\frac{1}{\sqrt N }\mathbb L} e^{-N(x-\log(1+x))}dx   $$
Since this set is contained within the set of infinitesimals, we can replace $x-\log(1+x)$ by its first order Taylor approximation and only do a small error again:
$$ N! \asymp N^{N+1} e^{-N} \int_{\frac{1}{\sqrt N }\mathbb L} e^{-\tfrac 12 Nx^2}dx $$
Now we can to a retransformation $x = \frac{1}{\sqrt N}y$ leading to
$$ N! \asymp N^{N} e^{-N} \sqrt N \int_{\mathbb L} e^{-\tfrac 12 y^2}dy  = \sqrt{2\pi N} N^N e^{-N} $$ 
which proves Stirlings formula as we have shown that for all unlimited $N$ there exists an infinitesimal $\epsilon$ such that 
$$ \frac{N!}{\sqrt{2\pi N} (N/e)^N} = 1 + \epsilon$$
