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I encountered the following change of summation $\sum_{i=1}^ \infty \sum_{j=1}^ \infty a_{i,j} = \sum_{j=1}^ \infty \sum_{i=1}^ \infty a_{i,j} $, in Ahtreya and Lahiri's "Measure theory and Probability theory", page 16, where it says that this is allowed since the summands $a_{i,j}$ are non-negative. I have encountered the same argument in other books of Measure theory, but never saw a proof in the appendix neither a reference of it.

From a web search I made, it seems that Fubini's theorem for double integrals in $R^n$, or, for a measurable function on a product measure space, is proved, and resembles very much my question. However, I cannot find a proof exactly for double sequences, as is my original question. This is quite strange, since this should be considerably easier to prove.

Can you kindly point me to one? Could you outline the proof, alternatively? Thanks a lot.

JamesPlayer
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    Some references are: Tom M. Apostol, Mathematical Analysis (2nd ed. 1974), Theorem 8.42; Alan F. Beardon, Limits: A New Approach to Real Analysis (1997), Theorem 5.5.3; John D. DePree & Charles W. Swartz, Introduction to Real Analysis (1988), Theorem 4.22 and Exercise 4.16; M. H. Protter & C. B. Morrey, A First Course in Real Analysis (2nd ed. 1991), Theorem 9.50; and in magisterial generality, J. Dieudonne, Foundations of Modern Analysis (rev. ed. 1969), (5.3.6) (but I long ago made a marginal note claiming to detect and fix an error in the proof, and I haven't checked it recently). – Calum Gilhooley Oct 24 '18 at 13:58
  • Wow! What an extensive list! Thank you so much! – JamesPlayer Oct 24 '18 at 14:37

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