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I am reading Singular Integrals and Differentiability Properties of Functions by E. M. Stein. In the second chapter the Calderón–Zygmund theory of singular integrals are extended to Hilbert space valued functions. However, the theory for integration of these vector-valued functions are only mentioned briefly. In particular, I have the following doubts:

  1. What is the difference between strongly measurable and weakly measurable, and is that a big problem (for investigations in harmonic analysis)?
  2. Why do we have $(L^p)'\cong L^{p'}$ in this case? The proof for $\mathbb{R}$ uses the Radon–Nikodym theorem, so probably we have to start from scratch to see that these are indeed true in the vector-valued case...

Thus my question is:

What is a book that proves all standard theorems on integration theory (especially $L^p$ spaces, Radon–Nikodym, etc.) in the context of vector-valued functions, preferably building the theory using the real theory (instead of starting from scratch), and does not get too technical?

I know of books (e.g., Real and Functional Analysis by S. Lang) which does cover these, but he starts from scratch. I wonder if there is a quick way to build the theory using the real case.

Thanks in advance!

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Vector Measures by Diestel and Uhl is the text that I learned from. I believe it checks the right boxes for you. It does get technical, however.

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