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Let $V$ be a (locally convex) vector space over $\mathbb R$, and $f$ a function $X\rightarrow V$. The goal is to integrate $f$ using the Pettis integral, i.e. finding $I\in V$ such that $$\psi(I) = \int_X\psi\circ f$$ for all continuous functionals $\psi$. Here $\int$ represents integration defined for functions $X\rightarrow\mathbb R$. A standard assumptions is that $X$ is a measure space and $\int$ is the Lebesgue integral. But what if a different notion of integration is used for $\int$, e.g. Daniell integration?

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    $\begingroup$ The Daniell integral depends on the choice of an initial vector lattice of functions equipped with a positive linear functional, the integral, which is then extended to a larger class of functions possessing similar properties compared to the Lebesgue integrable functions. So you need to specify your "initial data" before one can really answer your question. But notice that this initial choice often leads one back to ususl integrable functions for a given measure by a result due to Stone (see Royden, Real Analysis, 3.rd ed, Theorem 16.22). $\endgroup$
    – Ruy
    Jan 12, 2021 at 1:22
  • $\begingroup$ very nice comment. thank you very much $\endgroup$
    – lmaosome
    Jan 17, 2021 at 21:05

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