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Math and science::Analysis::Tao::08. Infinite sets

Fubini's theorem for infinite series

The extension of Fubini's theorem to infinite series is not a straightforward process. The proof acts as a summary of concepts including infinite series, series, limits, supremum and functions.

Let \( f: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N} \) be a function such that \( \sum_{(n,m) \in \mathbb{N} \times \mathbb{N}} f(n, m) \) is absolutely convergent. Then we have

\[ \begin{aligned} \sum_{n=0}^{\infty} \left( \sum_{m=0}^{\infty} f(n, m) \right) &= [...] \\ &= [...] \\ &= \sum_{m=0}^{\infty} \left( \sum_{n=0}^{\infty} f(n, m) \right)\end{aligned} \]

In other words, we can switch the order of infinite sums provided that the entire sum is absolutely convergent.