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Diclaimer: In what follows, I will be loose in the usage of terminology since the very nature of the question is of a similar flavour.

In the mathematics literature, one can find a zoo of theories of integration. Let $X$ be a 'nice enough' topological space (say, locally compact Hausdorff). To write an integral, I have encountered a few scenarios:

  1. There is a function on $X$ taking values in some topological vector space and a scalar-valued measure on $X$. This leads to a theory of integration for vector-valued functions. Example: Bochner integral, Gelfand-Pettis integrals.
  2. There is a real or complex-valued function on $X$ and measure on $X$ taking values in a topological vector space. This is the setting for vector measures and the theory of integration of scalar functions with respect to the vector measure. Example: Spectral integrals.
  3. There is a real or complex-valued function on $X$ and a measure on $X$ taking values in a monotone $\sigma$-complete partially ordered vector space. Example: The theories of integration due to J.D.M. Wright, P. K. Pavlakos.

There also theories of integration involving semigroup-valued measures and semigroup-valued functions.

As an analyst, I have found the formulations 1,2,3 to be quite useful (to say, not some idle generalizations). So I do understand the values of having integration theories in various contexts. But it also makes it easy to get lost while keeping track of these theories when an application comes due. I expect that some mathematicians must have undertaken the project of organizing them.

Question: Are there any survey articles which provide a organizational framework for the various theories of integration?

What do I mean by an organizational framework? In order to have a theory of integration, at the least, one needs functions on $X$ taking values in a ring $R$ and measures on $X$ taking values in an $R$-module $M$, or vice versa, so that the integral takes values in $M$. For issues of convergence, I expect one would have to impose topological or order-theoretic conditions on $R$ and $M$. One organizing principle could be the theorems we want in our theory. For example, if we want a version of Riesz representation theorem, what are the assumptions we should have on $R$ and $M$? Same goes for monotone convergence theorem, dominated convergence theorem.

Are there any survey articles taking the above approach?

Thank you.

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  • $\begingroup$ Integration in abstract spaces, T. H. Hildebrandt, Bull. Amer. Math. Soc. 59(2): 111-139 (March 1953), has a good survey on the state-of-the-art in 1953. $\endgroup$
    – genfuntranslate
    Commented Mar 19, 2021 at 7:37
  • $\begingroup$ In the paper, On integration in partially ordered groups by Pavlakos (Can. J. Math., Vol. XXXV, No. 2, 1983, pp. 353-372), the setting involves bi-additive functions $G_1 \times G_2 \to H$ where $G_1$ is a lattice-ordered group, and $G_2, H$ are partially ordered groups. So essentially, $G_1$-valued functions integrated against $G_2$-valued measures to yield an $H$-valued integral. $\endgroup$
    – genfuntranslate
    Commented Mar 19, 2021 at 7:38

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