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I'm reading Real Analysis: Modern Techniques and Their Applications and A Course on Abstract Harmonic Analysis, both by Folland.

First, a Radon (positive) measure is a Borel measure on a locally compact space $X$ such that:

  • $\mu(K) < +\infty$ $\forall K$ compact.
  • $\mu(A) = \inf\{\mu(U) : A \subseteq U \mbox{ and } U \mbox{ is open}\}$ $\forall A \in \mathcal{B}(X)$.
  • $\mu(U) = \sup\{\mu(K) : K \subseteq U \mbox{ and } K \mbox{ is compact}\}$ $\forall U$ open.

I have two questions:

In the first book, Fubini's Theorem is proofed for Radon (positive, I guess) measures. But I cannot find any reference for Radon complex measures. There exists a way to extend it to Radon complex measures?

Moreover, in the second book, Folland uses Fubini's Theorem for complex Radon measures in order to proof that convolution of complex Radon measures is associative, isn't it?

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$\int f \, d(\mu+i\nu) =\int f \, d\mu +i\int f \, d\nu$. There is no need for a separate Fubini's Theorem for complex measures.

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  • $\begingroup$ Thank you very much. Maybe my question is a bit stupid, and I think you're right, but it seems strange to me that I haven't found any reference in any book that says explictily what you say. So please, if you remember any, tell me. :) $\endgroup$ – Glazunov May 13 '18 at 14:54

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