# Proof of Fubini's Theorem for Series

I'm currently trying to self study Tao's Analysis I, and I'm really struggling with this proof - as I feel like I'm missing something. He goes really slow over sections I feel are obvious - for example he spends a while trying to get me to justify that for: $$f(x, y) \geq 0, \sum_{(x, y) \in A} f(x, y) \leq \sum_{(x, y) \in \mathbb{N} \times \mathbb{N}} f(x, y),$$ for $$A \subset \mathbb{N} \times \mathbb{N}$$ which seems intuitive to me. He suggests using a bijection to map it to $$\mathbb{N}$$ and I don't see why that's necessary. Therefore, I feel like I'm missing something critical in this section. I just started this book again after not reading it for a couple months - could this be why?

• The summand in the second sum is missing. Is it supposed to be $f(x,y)$? – saulspatz Jul 21 at 15:50
• @evaristegd thanks! – It'sNotALie. Jul 21 at 15:55
• @saulspatz yes, thank you :) – It'sNotALie. Jul 21 at 15:55

## 1 Answer

I don't have the book, so I don't know how Tao defines these sums, but I suppose it's something like the least upper bound of the set of all finite sums. Then, as you say, the theorem is intuitively clear, but that's not a proof. When you allow negative summands, you quickly get non-intuitive results. For example, a conditionally convergent sequence can be rearranged to converge to any given (finite or infinite) value. When the hypotheses of Fubini's theorem are not satisfied, you get non-intuitive results. (My intuition, at least is that Fubini's theorem for sums should always be true, but of course, that's wrong.)

Perhaps you see the need for proof, but think this particular proof is too obvious to be worth spending time on. Very well, just skim it and move on. On the other hand, if you think the proof is too long and involved, and can be much simplified, then you may well be misunderstanding something. If that's the case, you should post your simpler argument, and ask for verification. You'll either find your mistake, or have the gratification of knowing that you improved a proof by Terry Tao!

• Thanks a lot! I appreciate these words, it's the first time I self study so I'm nervous I'm making a mistake for retaining info. – It'sNotALie. Jul 21 at 19:30