TITLE: Is there something between summation and integration?
QUESTION [33 upvotes]: Let's take a general function $f(x)$, we can do a summation like:
$$\sum_{k=m}^n f(k)$$
And we can do an integration like:
$$\int_a^bf(k)dk$$
The basic difference between the two operation is that the first concerns only integer values of $k$ while the second is about real values of the function in an interval. So my question is: is there a similar operation that concerns rational values ? Something "between" summation and integration.

REPLY [2 votes]: Just to follow on from something inspired by john-mangual's answer:
Firstly one might know that the rationals $\mathbb{Q}$ are countable and thus can be enumerated by the positive integers $\mathbb{N}$. Let's pick an enumeration  $\mathbb{Q}=\{q_1,q_2,q_3,\ldots\}$. Now let $f:\Bbb R \rightarrow \Bbb R$ such that it is defined on $\mathbb{Q}$ (or restrict to a subset if necessary), then we can sum over the rationals by its enumeration, namely $$\sum\limits_{i=1}^\infty f(q_i).$$
Now unless the above summation converges absolutely, as in unless $$\sum\limits_{i=1}^\infty |f(q_i)| <\infty,$$
then the value of this series may depend on the enumeration of $\mathbb{Q}$, and even worse due to a theorem of Riemann for what are called conditionally convergent series, if you rearrange the terms you can make the series converge to whatever you like! 
However if it does converge absolutely then the sum is independent of how we choose to order the rationals, and we can sum in any order.
Maybe this will give you another insight into something else you can do, and since I don't really know any measure theory I can't help on that front!