- I saw $ \sum_{i \in \mathbb{Z}} \sum_{j \in \mathbb{Z}} f(i,j) = \sum_{k \in \mathbb{Z}} \sum_{(i,j) \in \{i+j =k, i, j \in \mathbb{Z}\}} f(i,j).$ I was wondering what conditions for it to hold are? Is there a name for the method using such equality?
I was also wondering if $\int_{x \in \mathbb{R}^n} \int_{y \in \mathbb{R}^n} f(x,y) du\, du = \int_{k \in \mathbb{R}^n} \int_{(x,y) \in \{x+y = k\}} f(x,y) du^2\, du$ may also be true? What are conditions for it to hold?
I am not sure if what I just wrote is mathematically correct. Especially, is the LHS two iterated integrals, each of which is an integral over $(\mathbb{R}, \mathcal{B}, u)$. Is the RHS also two iterated integrals, but the one inside is an integral over product Lebesgue space $(\mathbb{R}^2, \mathcal{B}^2, u^2)$?
Thanks!