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I have a question about the definition of the integral in nonstandard analysis. The definition that I've usually seen is this: given a function $f(x)$ that you want to integrate from $a$ to $b$, you define a function $$S(\Delta x) = \sum_{x = a}^b f(x) \Delta x,$$ which you then extend to a hyperreal function through the transfer principle. Then the integral is defined as the standard part of $S(dx)$, where $dx$ is infinitesimal. The definition feels fine to me except that it looks like it is equivalent to defining an integral through left sums, which is famously not the same as a Riemann integral. So my question is what's wrong here? Is this definition that I gave not the usual one used in nonstandard analysis? Is my argument that this is equivalent to left sums flawed? Is the nonstandard integral actually different than the Riemann integral?

sudgy
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  • This needs more detail to narrow the scope. In particular: Where have you seen this definition? What was $\displaystyle \Sigma_{x=a}^b$? Was $f$ assumed to be bounded/continuous? What did $x$ range over? Was $dx$ a fixed infinitesimal, or was the integral defined as the common standard part as $dx$ varies over all infinitesimals (assuming that this standard part exists)? Without more information, this question essentially turns into "compare all possible nonstandard ways of defining the integral to the Riemann integral". – Z. A. K. Sep 20 '22 at 09:29
  • I first came across this definition in https://math.stackexchange.com/questions/464565/definition-of-the-integral-in-non-standard-calculus, while I've also seen a textbook about it in https://people.math.wisc.edu/~keisler/chapter_4.pdf. – sudgy Sep 20 '22 at 16:03
  • Keisler's chapter starts with "a real function $f$ continuous on the interval $I$", and then defines the integral over proper subintervals of $I$. If I recall correctly, left-sum Riemann integrability does coincide with the usual (tagged) notion of Riemann integrability for such functions. – Z. A. K. Sep 20 '22 at 21:02
  • Ah, that would explain it. Although now I'm left wondering how you would define the Riemann integral in nonstandard analysis in the non-continuous case, but that's a different question. I guess I'll answer this myself. – sudgy Sep 21 '22 at 02:59
  • A modern/contemporary definition: $\int_I f(x) dx$ is the common standard part of the hyperfinite sums $\Sigma_{x \in H} f(x) \Delta_H(x)$ as $H$ ranges over all hyperfinite sets $H$ satisfying $I \subseteq H\subseteq ~^\star I$, and $\Delta_H(x)$ is the distance of $x \in H$ from the closest $y \in H$ so that $x \neq y$. The integrability condition is then just the existence of this common standard part. – Z. A. K. Sep 21 '22 at 03:24

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As mentioned in a comment, this definition of the nonstandard integral is only used for continuous functions, in which case the integral based on left sums is the same as the Riemann integral.

sudgy
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