Definition of the integral in non-standard calculus

Question

In the sources I've seen, the integral is defined in non-standard calculus as the hyperreal extension of a function related to Riemann sums. E.g.,

Let $$ S(\Delta x) = \sum_{a}^{b}f(x)\Delta x$$ be a function of a Riemann sum where $\Delta x$ is the length of each partition. Then let $S^*(\Delta x)$ be the hyperreal extension of that function, which exists due to the transfer principle. Then we can talk about $S^*(\rm dx)$, where $\rm dx$ is an infinitesimal.

This is very simple and seems to work, but it doesn't really explain what's happening in the function $S^*(\Delta x)$. What does a Riemann sum with infinitesimal partitions mean? How do I sum $H$ terms, where $H$ is a hyperinteger?

Can anyone direct me to other definitions of integrals in non-standard calculus (not necessarily Riemann equivalent) that answer some of these questions? Or possibly a textbook that goes into more detail about this subject?

Edit: Let me clarify something. I understand the definition. I certainly don't think it's not well-defined.

I just feel it sort of "covers up" the really interesting bits. "Interesting bits" includes, for example, using induction on the hyperintegers.

Answer 1

The integral is not the hyperfinite sum $S^{\ast}(dx)$ but rather the standard part of that hyperfinite sum. This is similar to the situation with the derivative, where one can form the quotient of infinitesimals $\frac{dy}{dx}$ where $dy$ is the $y$-increment corresponding to the $x$-increment $dx$. Here the derivative is not the quotient but rather the standard part of that quotient. Usually one denotes the infinitesimal $x$-increment by the symbol $\Delta x$ and the corresponding $y$-increment by $\Delta y$. The derivative is then defined to be $$\text{st}\left(\frac{\Delta y}{\Delta x}\right)$$ rather than the ratio itself. Here "st" is the standard part function. Then one sets $dx=\Delta x$ and defines $dy$ by $dy=f'(x)dx$ so that the derivative can be written as $dy/dx$.

I have had good experience teaching using Keisler's textbook which is available online here. A more advanced companion volume by Keisler that covers more of the theoretical material can be found here.

To answer more specifically your questions "What does a Riemann sum with infinitesimal partitions mean? How do I sum H terms, where H is a hyperinteger?": your intuition of a very small partition step and a very large number of partition points is already the right intuition. The notion of an infinite hyperinteger is merely a way of formalizing your intuition. The advantage of the hyperreal approach is that such intuitions, which remain at the level of vague musings in the epsilon, delta approach (without infinitesimals), can now be formalized rigorously. This allows one to write down proofs that are closer to the original intuition than in the epsilon, delta approach. One notion that gets clarified this way is that of a Cauchy sequence; see the related discussion I would like to know an intuitive way to understand a Cauchy sequence and the Cauchy criterion.

Answer 2

Due to the transfer principle, as long as you don't compare them against each other, nonstandard analysis is completely indistinguishable from standard analysis.

In standard analysis, you have the (standard) finite summation operator: given (standard) integer limits $a \leq b$ and a (standard) real-valued function $f$ of the integers, the operator produces a real number $S$, which we often write as

$$ S = \sum_{n=a}^b f(n) $$

If we apply transfer, we get the (nonstandard) hyperfinite summation operator: given hyperinteger limits $a \leq b$ and an internal hyperreal-valued function $f$ of the hyperintegers, the operator produces a hyperreal number $S$. We might write it as

$$ S = {}^\star\sum_{n=a}^b f(n) $$

to emphasize that this is the nonstandard summation operator rather than the standard summation, but I would prefer to just write it as

$$ S = \sum_{n=a}^b f(n)$$

If you were so inclined, you don't even have to bother with the transfer principle. The nonstandard integers satisfy the (internal) principle of induction, so you could just recursively define the nonstandard summation operator in any usual way, such as

$$ {}^\star\sum_{n=a}^a f(n) = f(a) $$ $$ {}^\star\sum_{n=a}^{b+1} f(n) = f(b+1) + {}^\star\sum_{n=a}^b f(n) $$

which recursively defines the summation for all hyperintegers $a,b$ with $a \leq b$, and for all (internal) hyperreal-valued functions $f$ of the hyperintegers.

The reason you are having trouble seeing that the summation is well-defined is because you are implicitly trying to use induction over standard integers to justify recursions over nonstandard integers, but that simply doesn't work.