# What is the name of this method in Lebesgue integral

1. 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?
2. 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!

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Part 2 of your post needs some heavy rewriting but both parts 1 and 2 are concerned with Fubini theorem. The net result is that if one series/integral converges for $|f|$, or if $f\ge0$, you can change the order of summation/integration and/or regroup things at will, the result will always exist and be the same no matter what ordering and regrouping method you used.