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May 2 at 12:49 vote accept G. Blaickner
May 2 at 12:18 answer added Piero D'Ancona timeline score: 3
May 2 at 11:46 comment added Hannes This related question about the theorem in Diestel-Uhl for $\sigma$-finite measure spaces popped up in my sidebar. Also, regarding your follow-up questions, for $p=1$ and $\Omega$ an interval, the Diesel-Uhl result also holds true for any Banach space with the RNP if I see it correctly.
May 2 at 10:51 comment added gerw If I remember correctly, something can be found in the book "Functional Analysis: Theory and Applications" by Robert E. Edwards. However, I do not have a copy at hand and I do not remember exactly which results are contained there.
May 2 at 8:49 history asked G. Blaickner CC BY-SA 4.0