# Can one define the Lebesgue integral with nonstandard analysis?

Reading Keisler's books about non-standard analysis, I have begun to suspect the possibility that there may not be an adequate characterization of the Lebesgue integral and of measure theory in general using non-standard analysis.

Question: Is this an actual weakness of non-standard analysis? Or is Keisler not doing the subject justice by failing to mention that the Lebesgue integral and its generalizations can also be well understood in the framework of nonstandard analysis?

Explaining in detail how the construction of the Lebesgue integral carries over to non-standard analysis would be too broad and too much to ask or expect of any answer, so I will without hesitation accept answers only giving references to sources which discuss this issue. This is why I have tagged the question (reference-request).

Background: I have been reading Keisler's books about non-standard analysis, the introductory Calculus: An Infinitesimal Approach and the more advanced and rigorous Foundations of Infinitesimal Calculus. The argument for why differentiation is more intuitive with non-standard analysis is fairly clear to me, however, I am not convinced yet that the same is true of integration.

Specifically, both of Keisler's books only mention how to perform Riemann integration using non-standard analysis, but say nothing about Lebesgue integration (i.e. general integration with respect to an arbitrary measure, not just the Lebesgue measure on $\mathbb{R}$ and its subsets). In fact, the Lebesgue integral is not mentioned at all. To me at least, this seems like an unusually large omission, at least from Foundations of Infinitesimal Calculus, which seems to be fairly rigorous.

• It’s a perfectly natural omission from Foundations: Foundations was always intended to be the rigorous companion to Calculus, filling in the background for instructors or advanced undergraduates. Since the Lebesgue integral is not part of a standard calculus course, there’s no reason to treat it in Foundations. A little searching online will quickly turn up non-standard treatments of various measures and generalized integrals. – Brian M. Scott Sep 2 '16 at 19:32
• It's also worth noting that Lebesgue's approach isn't the last word on integration; see the generalized Riemann integral. – user14972 Sep 4 '16 at 19:42

Thinking about how to answer the title question... I think there is actually a fairly straightforward path.

By the regularity theorem, if $X$ is a standard Lebesgue measurable set and we pick a positive infinitesimal $\epsilon$, there exists a closed set $C$ and an open set $U$ such that:

• $C \subseteq X \subseteq U$
• $\mathop{\mathrm{std}}(\mu(U)) = \mu(X)$
• $\mathop{\mathrm{std}}(\mu(U \setminus C)) = 0$

This now makes it obvious how to define the Lebesgue measure.

Every open subset of the reals is a disjoint union of open intervals; let $\lambda$ be the function computing the total length. Then,

A standard set $X$ is said to be Lebesgue measurable if and only if there exists a closed set $C$ and an open set $U$ such that

• $C \subseteq X \subseteq U$
• $\lambda(U \setminus C) \approx 0$

The Lebesgue measure of such a set is defined to be $\mu(X) = \mathop{\mathrm{std}}(\lambda(U))$

(note that the standard part of a positive infinite hyperreal is the extended real $+\infty$)

This can be viewed as an expression of Littlewood's first principle:

Every measurable set is nearly a finite sum of intervals

I imagine you can even restrict the $U$ above to be a (hyper)finite sum of open intervals.

I'm too tired to continue, but once we have the Lebesgue measure, I can't imagine there being any difficulty in defining the integral by approximating a standard function by a nonstandardard simple function.

• I should disclaim that I have no reason to believe this is the best approach to the subject. (or even that developing the Lebesgue integral is really the way you want to go when using NSA) – user14972 Sep 12 '16 at 8:10
• This is really nice though, and arguably makes the regularity theorem easier to understand/think about -- same thing with sets of infinite measure. I appreciate the thought you've put into this. – Chill2Macht Sep 12 '16 at 20:01
• Hurkyl, this is not the traditional approach to the Lebesgue integral via Robinson's framework and I think your approach is more complicated. Would you like to mention something about the traditional approach? – Mikhail Katz Oct 10 '16 at 7:53
• @MikhailKatz: My google-fu had been unable to find any mention of a traditional approach. If there is one, by all means provide it as an answer! – user14972 Oct 10 '16 at 8:27
• The what is by now the traditional approach was pioneered by Loeb in the 1970s and passes via the Loeb measure. The idea is that everything can be gotten out of a counting measure but the counting has to be hyperfinite. In a 0th approximation, you approximate your set by an internal one to which a hyperfinite count can be applied. Goldblatt has a nice treatment though it could perhaps be simplified. – Mikhail Katz Oct 10 '16 at 8:35