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Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, parameterized by a compact set $E\subset \mathbb R^N$. Let's say the map $t\mapsto \omega_t$ is smooth.

Then is there a natural way to talk about the 'average' or 'integration' of this family? The 'average' of this family should be thought of $$ \frac{1}{m(E)}\int_E\omega_t dt $$ It may be similar to Bochner integrals for Banach-space-valued functions, but $\Omega^*(X)$ is not Banach. Is there a more general integration theory to deal with this case?

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    $\begingroup$ For each $x$ in $X$, $\omega _t(x)$ lives in a finite-dimensional vector space, you can just integrate it. Clearly you get a smooth differential form. $\endgroup$
    – abx
    Commented Nov 3, 2018 at 18:23
  • $\begingroup$ @abx Thanks for the comment. So, if $\omega$ is the pointwise integration as you said, then can we guarantee it is still smooth, namely, in $\Omega^*(X)$? I guess the same thing should also work if we replace $\Omega^*(X)$ by some general thing. $\endgroup$
    – Hang
    Commented Nov 3, 2018 at 18:27
  • $\begingroup$ Yes for $\Omega ^*(X)$ — just use local coordinates. For "more general things", well, it depends on what things... $\endgroup$
    – abx
    Commented Nov 3, 2018 at 19:21
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    $\begingroup$ It’s really no different from integrating a function over a domain in $\mathbb{R}^n$, where the function depends on a finite number of parameters. In fact, if you write $\omega$ with respect to local coordinates, that’s exactly what your integral is. $\endgroup$
    – Deane Yang
    Commented Nov 3, 2018 at 19:24

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