I am aware of three types of summations:
The discrete sum can be used on a finite or countably infinite set, such as the naturals.
$$ \sum_{n \in \mathbb{N}} \frac{1}{2^n} $$
The integral sum can be used on an uncountable set provide that such set are continuous. For example on the reals:
$$ \int_a^b x dx $$
Finally, I am aware of the functional integral which sums over an uncountable space of functionals:
$$ I=\iint D\gamma \exp \left(\int_a^b Ldt \right) $$
However, in the case of the powerset of $\mathbb{N}$, I obtain uncountably infinitely may set of natural numbers (including infinite sets). What type of sums, notations and what is the concept behind summing over such a set? I do not see how any of these three sums are applicable. For instance, the set of sets does not appear to work well with integrals. How can I then do this?
Specifically, I have a function $F:\mathcal{P}[\mathbb{N}] \to [0,1]$ and I would like to sum it over all elements of the power set of the natural numbers $\mathbb{N}$? You can think of it as normalization. I assume can't use integrals, because I can't define an infinitesimal of $\mathcal{P}[\mathbb{D}]$.
You want $$ \sum_{x \in \mathcal{P}(\Bbb{N})} F(x) $$
In real analysis, the sum of a set of nonnegative summands is defined to be the supremum of the collection of sums of finitely many summands. (Since the summands are nonnegative, we may rearrange them at will and get the same result. Contrast with the Riemann Rearrangement Theorem.)
Note that if $F$ takes values greater than any particular positive value infinitely many times, the finite sets that include more and more of these lower bounded summands will force the supremum to infinity. So to have a finite sum, the number of $F$ values greater than any particular lower bound must be finite. Then, much as the terms of a (usual) series must go to zero, the infimum of the summands of this sum must be zero. (The easiest way for this to happen is for only finitely many summands to be nonzero.)