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I've been studying dimensional regularization recently and found that some people view the divergent integrals, which are replaced by certain finite expressions in the regularization procedure, as existent because "the regularization process provides the correct understanding what the integral really is". Thus they see those expressions no longer as Lebesgue integrals but as objects of a kind for which they don't have a rigorous mathematical definition. I'm looking for such definitions. This has parallels in the study of divergent series, which may be Cesàro-summable for example. On the other hand, if one views series as special cases of Lebesgue integrals with (counting measure), only absolutely convergent series exist in Lebesgue sense. We thus have a vast theory of generalizations to Lebesgue integrals with counting measure. An often quoted reference seems to be Hardy's book "Divergent Series" although I'd be interested in more modern sources, as well as for asymptotic analysis! Apart from looking for such references,the question is:

Are there generalizations of integrals over, say, $\mathbb{R}^n $ that (for example) do not require the absolute value of the integrand to be integrable as well? Generalizations in the style of Cesàro? Do such generalizations have something to do with regularization schemes? Are there connections to topics like distribution theory?

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    $\begingroup$ Usually there exists a rigorous interpretation given by an analytic continuation of the integrand by including one or more parameters. $\endgroup$ Jul 8, 2017 at 22:17
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    $\begingroup$ I'm looking for general definitions including specifying what functions are integrable (which should preferably be a vectorspace) and what their integrals are, which extends Lebesgue integration. If there is a treatment of the usual analytic continuation techniques in such a general style, rather than just being concerned with individual Feynman diagrams, I would love a reference. $\endgroup$ Jul 8, 2017 at 22:34
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    $\begingroup$ the Denjoy integral is one example for your first question $\endgroup$
    – user363464
    Jul 9, 2017 at 7:16
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    $\begingroup$ I mean, Cesàro is quite easy to apply. Consider a regularization of the form$$\int_a^\infty f(x)~\mathrm dx=\lim_{n\to\infty}\frac1n\sum_{k=1}^n\int_a^{a+k}f(x)~\mathrm dx$$And of course, extend to multiple variables as needed in whatever manner you are interested in. $\endgroup$ Aug 15, 2017 at 21:27
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    $\begingroup$ This is related: mathoverflow.net/questions/115743/an-algebra-of-integrals/… $\endgroup$
    – Anixx
    Jan 11, 2020 at 15:25

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