Rudin book's, Real and complex analysis chapter 6.
Of (1) every rearrangement of series must also converge.
My questiion is: Why concludes that $\mu(E)$ converges absolutely?
Thank you all.
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It would help if we knew what Theorem 3.56 was. – Math1000 Dec 30 '14 at 01:54
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Theorem 3.56: If $\sum a_n$ is a series of complex numbers which converges absolutely then every rearragement of $\sum a_n$ converges, and they all converge to the same sum. – user126033 Dec 30 '14 at 02:01
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Theore 3.56 that you've cited seems to prove another direction. Perhaps, here we rather shall use Riemann rearrangement theorem. Notice that in your case, not only every rearrangement converges, they all do converge to the same limit $\mu(E)$. If the series were not absolutely convergent, then there would exist two rearrangements with different limits.
