TITLE: Definition of the integral in non-standard calculus
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.