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I have a Fréchet space $V$ whose topology is (if it helps) induced by a family $\mathcal{P}$ of norms - not just seminorms - and on this space I have a Borel probability measure $\nu$. Now, I would like to see whether it is possible to make sense of the integral

\begin{equation} \int_V x \, \mathrm{d} \nu \left( x \right) \end{equation}

within $V$. The measure is such that I actually can prove that the integral exists as a Bochner integral in some Banach space completions of $V$ with respect to some of the norms in $\mathcal{P}$. But I am far away from being able to prove this for all of these norms.

Is there perhaps a different way to define Bochner integrals in Fréchet spaces? I only know the usual Banach space setting and it seems to not be enough in this case.

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  • $\begingroup$ The following paper might be useful: G. Erik F. Thomas. Integration of functions with values in locally convex Suslin spaces. Trans. Amer. Math. Soc., 212:61–81, 1975 $\endgroup$
    – Christian
    Commented Jan 14, 2021 at 14:02
  • $\begingroup$ I think in your setting you only need that the integral $\int_V p(x)\,\mathrm{d}\nu(x)$ is finite for every $p\in\mathcal{P}$ - then the results in the above mentioned paper should give you that the integral is an element of $V$. $\endgroup$
    – Christian
    Commented Jan 14, 2021 at 14:29
  • $\begingroup$ I just encountered the same theorem in the paper. I am checking the details, but it seems to be correct :) $\endgroup$
    – iolo
    Commented Jan 14, 2021 at 14:44
  • $\begingroup$ Feel free to write an answer btw $\endgroup$
    – iolo
    Commented Jan 14, 2021 at 14:56

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