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Maybe we can restrict the discussion to one dimension:

When can we integrate a distribution over an interval?

In Lebesgue theory, we can integrate measurable functions. But distributions are not functions. They can be integrated against smooth test functions. So my question is: when can we actually take the test function as an indicator function for an interval?

For instance, what is the integral of Dirac at $0$ over $[0,1]$ or $(0,1)$?

Can we integrate something from $H^{-1}$ (Sobolev space) over $[0,1]$?

Another example of a (random) distribution is Gaussian white noise. Integrating it over a domain $A$ gives a mean zero Gaussian random variable, whose variance is the area of $A$. Note that Gaussian white noise is not measurable function...

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    $\begingroup$ In general, distributions are not integrable. You can inject the space of all locally integrable functions on the space of distributions and these are all integrable distributions you are going to get. You can interpret some distributions as measures and integrate things against them. $\endgroup$
    – Max Reinhold Jahnke
    Commented Oct 27, 2017 at 2:21
  • $\begingroup$ Looking at $\lim_{\epsilon \to 0}\langle T, \psi \ast\phi_\epsilon \rangle$ for some mollifier $\phi_\epsilon$, then integrating with respect to $\psi$ a compactly supported piecewise $C^\infty$ function makes sense iff around the boundary the distribution $T$ is of order $0$. If you want continuity wrt to deformation of the mollifier and the boundary then you need a little more, see $\langle \delta, 1_{x > a}\rangle$ which is a good example. If $S$ is a compactly supported distribution then $S\ast T$ makes sense as a distribution $\langle S\ast T,\varphi\rangle=\langle T,S\ast\varphi\rangle$ $\endgroup$
    – reuns
    Commented Oct 27, 2017 at 2:27

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A simple and elementary theory of integrals of distributions was developed by several authors in the $50$'s and $60$'s. Since every distribuion has a primitive, this reduces to defining the limit of a distribtution at a point. Thus in the two examples you mention, the integrals are, as one would expect, $1$ and $0$ respectively. Perhaps the simplest version of this theory is due to the portuguese mathematician J. Sebastião e Silva---it can found in his original papers (e.g., "On Integrals and Orders of Growth of Distributions") or in the book "Theory of Distributions" by Campos Ferreira.

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  • $\begingroup$ Can you give exact references? For example, Campos Ferreira is not found in zbMath zbmath.org/authors/?s=0&q=Ferreira%2CC. $\endgroup$
    – user64494
    Commented Oct 27, 2017 at 12:39
  • $\begingroup$ There is a site in Portugal with all of Silva's papers (sebastiaosilva100anos.org). You can go straight to the relevant article by googling the title in my answer and should be able to find the book at mathscinet---if I remember rightly, it was published by Pitman. $\endgroup$
    – molendinar
    Commented Oct 27, 2017 at 15:24
  • $\begingroup$ Have now checked---the book's review at Math. Review is MR 1477907 $\endgroup$
    – molendinar
    Commented Oct 27, 2017 at 15:29
  • $\begingroup$ The link sebastiaosilva100anos.org is dead. $\endgroup$
    – user64494
    Commented Oct 27, 2017 at 15:44
  • $\begingroup$ Yes, indeed Ferreira, J. Campos. Introduction to the theory of distributions. Translated from the 1993 Portuguese original by J. Sousa Pinto and R. F. Hoskins. Pitman Monographs and Surveys in Pure and Applied Mathematics, 87. Longman, Harlow, 1997.$ {\rm xiv}$+157 pp. ISBN: 0-582-31144-6 MR1477907. Can you quote the results you are refer to? $\endgroup$
    – user64494
    Commented Oct 27, 2017 at 15:57

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