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Ever since the development of Theory of integration. each specific problem has led to development of new integrals.

  1. 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.

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

  3. 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?

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  • $\begingroup$ Rough paths are a generalization of Riemann Stieltjes integration and regularity structures are a generalization of those. Regularity structures are a very active and modern area of research. I'm not sure if you can really call those integration however. $\endgroup$ – user223391 Jan 5 '18 at 1:25
  • $\begingroup$ Are you talking only about integration of a function of a real variable? If not, I would say: the Control Measure Problem. $\endgroup$ – GEdgar Jan 5 '18 at 1:25
  • $\begingroup$ @GEdgar No, I am talking in general. But since these integrals are well known, I only cared about writing those. $\endgroup$ – yasir Jan 5 '18 at 1:28
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    $\begingroup$ Here are some problems in rough paths. sites.google.com/site/horatioboedihardjo $\endgroup$ – user223391 Jan 5 '18 at 1:29
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    $\begingroup$ See if problem 1.33 helps you. $\endgroup$ – Rohan Jan 7 '18 at 10:26
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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

$$ \frac{N!}{\sqrt{2\pi N} (N/e)^N} = 1 + \epsilon$$

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  • $\begingroup$ send me a link to your Master Thesis. now. $\endgroup$ – user394255 Jan 7 '18 at 21:31
  • $\begingroup$ @Hyperplane Looks like a great idea. I'd love to take a look at your thesis to get started. Will that be possible for you? $\endgroup$ – yasir Jan 9 '18 at 0:56
  • $\begingroup$ @Adrianos Sorry I have been very busy the last week and kinda forgot about this. I don't feel super comfortable publishing my master thesis online, but I would be willing to send you a copy via e-mail. I added an address to my profile which you can use to reach me. Although to be honest my thesis only touches on this issue of how get around defining this sort of integration. In the end as time ran short, and the main goal was sth. completely different, my supervisor and I decided it would be ok to continue axiomatically from some point on. $\endgroup$ – Hyperplane Jan 19 '18 at 0:20
  • $\begingroup$ @yasir Sorry I have been very busy the last week and kinda forgot about this. I don't feel super comfortable publishing my master thesis online, but I would be willing to send you a copy via e-mail. I added an address to my profile which you can use to reach me. Although to be honest my thesis only touches on this issue of how get around defining this sort of integration. In the end as time ran short, and the main goal was sth. completely different, my supervisor and I decided it would be ok to continue axiomatically from some point on. $\endgroup$ – Hyperplane Jan 19 '18 at 0:20
  • $\begingroup$ @Hyperplane I am not able to contact you from the address attached to your profile. Would you mind dropping me an e-mail at ykhan9501@gmail.com $\endgroup$ – yasir Feb 5 '18 at 14:02

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