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Let $E$ be a locally convex Hausdorff space, and $X$ be a locally compact Hausdorff space which we fix a positive Radon measure $\mu$. Assume that $f: X \to E$ is a function such that the Pettis-integral $$\int_X f d \mu$$ exists.

Let $p$ be a continuous seminorm on $E$. Is it true that the inequality $$p\left(\int_X f d\mu\right)\le \int_X (p\circ f) d\mu$$ holds?

Context: This seems to be implicitly used in the proof of lemma 2.4 chapter VI "Left Hilbert Algebras" in Takesaki's second book "Theory of operator algebras".

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This is almost correct. Note that $p \circ f$ might not be measurable, which is why the right-hand side has to be replaced by the lower Lebesgue integral. The result then follows from the Hahn-Banach theorem, see Wikipedia for the argument and the definition of the lower Lebesgue integral.

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  • $\begingroup$ Thanks! Splendid answer! For my purposes, I can assume $f$ to be continuous, so no problem with measurability then, I think. $\endgroup$
    – Andromeda
    Commented Mar 22, 2023 at 21:53
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    $\begingroup$ Glad this helped. Basically, I just copied the Wikipedia paragraph.. :-D $\endgroup$ Commented Mar 22, 2023 at 21:54
  • $\begingroup$ Somehow, I read over that part :D $\endgroup$
    – Andromeda
    Commented Mar 22, 2023 at 21:55

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