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In the Notes section of chapter 2 of Diestel and Uhl's Vector Measures they make the comment:

"Presently the Pettis integral has very few applications. But our prediction is that when (and if) the general Pettis integral is understood it will pay off in deep applications." ( Google books link )

That was back in $1977$ and when I did a little searching to find out how well that prediction had fared I found little: wikipedia provides a cursory description and Encyclopedia of Mathematics is arguably better but terser. This question asks why (Dunford-)Pettis integrals are useful, but I would not say it covers deep applications per se.

So: are there deep applications to the Pettis integral, and if so, what are they?

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Diestel and Uhl commented on this in Measure theory and its applications:

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  • $\begingroup$ That's excellent, thank-you! $\endgroup$
    – postmortes
    Commented Jul 30, 2020 at 11:28

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