# Spaces of functions on subsets and linear mappings

I have been looking for research concerned with the following construct: Let $X$ be a set (most likely with a topology) and let $R$ be a subset of the family of all subsets of $X$. $R$ could be the power set of $X$, or a ring, or a Borel family, etc. of subsets. I am interested in the space of all real-valued functions on $R$ (call it $L$) and further, the subspace of $L$ consisting of those functions that are additive on disjoint elements (I realize there are issues of countably additive vs. finitely additive, etc., but this is the general idea).

I was wondering if anyone had seen any work in this kind of context and might offer some "keywords" so I could track down papers, parts of books, etc. to research my issue.

Thanks much for any information.

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Did the term "set-functions" give you interesting things? –  Davide Giraudo Mar 7 '12 at 16:13
If you have access to MathSciNet, the exact phrase "Additive Set Functions" should return a lot of results, including the classic paper of Bochner and Phillips Additive Set Functions and Vector Lattices. Citations to it and citations from it will probably get you started. –  Willie Wong Mar 7 '12 at 16:45