"Non countable" Series or products Consider $$\sum_{i \in \mathbb R}i$$ Or something like $$\prod_{i\in\mathbb R^*} i$$
Intuitively, one might say in the first case, since every $i$ has an additive inverse, the sum would be $0$. In the second case, similarly, since every $i \in \mathbb R^*$ has a multiplicative inverse, then the product would be 1. ¿Is there any theory regarding this kind of "series" or products? More generaly, one might consider an operation "$\odot$" defined on a family of sets $A_i$ where $i \in I$ and I is non countable, and consider $$\bigodot_{i \in I} A_i $$ 
I'm just asking out of curiosity
 A: Disclaimer: I haven't taken real analysis (yet), but here is my intuition.
One can make arguments like these, but there is still the problem that these sums and products diverge. For example, even the sum
$$\sum_{n=0}^{\infty}(-1)^n \left \lceil{x}\right \rceil $$ diverges, as we can see that the partial sums will alternate between 0 and a high number. Even though we can pair up each terms to get
$$\sum_{n=0}^{\infty}0$$
We cannot say that this sum converges to 0.
Thus I do not believe that we can say what these sums or products are equal to, since they diverge anyways.
A: There is such a thing as a "Cauchy principal value" of an integral, which, as applied to sums rather than integrals, could be something like $\displaystyle\lim_{N\to\infty} \sum_{n=-N}^N a_n.$
Notice that if you take $\displaystyle\lim_{N\to\infty} \sum_{n=-N}^{2N} n,$ you will get a different result from what you get with $\displaystyle\lim_{N\to\infty} \sum_{n=-N}^N n$, even though both sums have the same set of terms. That sort of thing can happen only if the sum of the positive terms and the sum of the negative terms both diverge to infinity.
