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
\[
\begin{aligned}
\sum_{n=0}^{\infty} \left( \sum_{m=0}^{\infty} f(n, m) \right) &= \sum_{(n,m) \in \mathbb{N} \times \mathbb{N} } f(n, m) \\
&= \sum_{(m,n) \in \mathbb{N} \times \mathbb{N} } f(n, m) \\
&= \sum_{m=0}^{\infty} \left( \sum_{n=0}^{\infty} f(n, m) \right)\end{aligned} \]
In other words, we can switch the order of infinite sums provided that the entire sum is absolutely convergent.
If you have time, it's worth working through the proof.