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I have been studying measure theory and it has occurred to me that so far I have not developed the intuition behind an abstract integral. I believe I understand what it is in technical terms, but I'm trying to develop a more formal intuition.

We may interpret the usual Riemann integral on $\mathbb{R}^n$ as the area under the curve obtained by partitioning the domain into finer and finer partitions. Likewise, the Lebesgue integral on $\mathbb{R}^n$ can be interpreted as also being the area under the curve, but this time partitioning the range of the function and taking the Lebesgue measure on the preimage of these range sets.

However, what alludes me is what does integration represent in a general sense? Does such a geometric interpretation even exist in the general case, or do we simply take it for what it is (i.e. a map from some measurable space $X \rightarrow \mathbb{R}$?

CBBAM
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    an abstract integral is, in general, a positive linear functional over a set of functions – Masacroso May 21 '22 at 04:11
  • @Masacroso Is there no interpretation beyond it being a linear functional? – CBBAM May 21 '22 at 04:12
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    There are many interpretations, as you seem to know, more specific than "being a linear functional". However it is reasonable to start with a quite general setting and look for interpretations of intermediate specificity between that and the Riemann/Lebesgue integrals. I'm fond of the expected value interpretation from probability. Your studies will no doubt expose you to many such interpretations, and you may enjoy organizing them in to an outline of graded generality. – hardmath May 21 '22 at 04:46
  • See the answers to this question – Jean Marie May 21 '22 at 04:50
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    For many integrals, interpretation in terms of area is not adequate (such as in line integrals, complex integrals, abstract Lebesgue integrals, operator integrals, etc.) I prefer to regard integrals as weighted sums. – Sangchul Lee May 21 '22 at 06:03

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