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In measure theory one usually starts with a $\sigma$-algebra $A$ of sets and considers a measure $\mu:A\to [0,\infty]$. I'm interested in abstracting this definition to allow more general domains and codomains for the measure function. I'm aware of measures allowed to take values in $\mathbb {R}$ as well as in $\mathbb {C}$ and I'm also aware of vector valued measures. I'm not really familiar with measures having a domain that is anything but a $\sigma$-algebra of sets (except of course for finitely additive measures).

So, any information or reference to work done along these lines would be greatly appreciated. Especially, for the most general case where the domain of $\mu$ is allowed to be any $\sigma$-complete lattice and the codomain is the most general kind of lattice (probably complete with some binary operation $+$) that will support a good theory. But also non so far-reaching generalizations would be great. Thanks.

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You can read Rudin's Real and Complex Analysis. The first 6 chapters answer all of the questions. –  Hui Yu Nov 5 '12 at 2:32
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@ Hui Yu, I have that book in front of me and I can't find any reference (not in the first six chapters nor anywhere else) to measures whose domain is not a \sigma algebra of sets nor codomain being substantially different than the complex numbers. Can you perhaps point me to a more precise location in the book? –  Ittay Weiss Nov 5 '12 at 3:10
    
I don't have the book with me. But I vaguely recall that in the first/ second chapter Rudin define measure spaces where the codomain is a general topological space. As for the domain, I don't think we can do without a sigma algebra. Otherwise, we do not have countable summability of measures. –  Hui Yu Nov 5 '12 at 3:40
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I don't understand how the codomain can be taken to be an arbitrary topological space since then you can't expressed even finite summability. As for the domain, I don't see why it has to be an algebra of sets. Why not just a complete or sigma-complete lattice? –  Ittay Weiss Nov 5 '12 at 4:37
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Tweaking the domain is more of a curiosity at this point. Tweaking the codomain is more interesting to me and can be explained in two ways. Just like considering signed and complex valued measures is important there may be other useful codomains for measures that will be relevant for real analysis. Another reason is that if one can have a nice measure theory in the more general setting than perhaps useful codomains can be found to measure size of sets by things different than real numbers (in particular I'm thinking of the Levy space as a codomain of interest). –  Ittay Weiss Nov 5 '12 at 5:08

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Heinz König did a lot of work on measures defined on lattices and he has written a very demanding book on these issues, Measure and Integration: An Advanced Course in Basic Procedures and Applications. Ultimately, one can extend these measures to $\sigma$-algebras though, König is mainly concerned with extension procedures and regularity properties as far as I can tell.

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Thank you Michael, great reference exactly along the lines I was looking for. –  Ittay Weiss Nov 5 '12 at 20:46

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