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Are there algebraic structures that integrate groups with probability measures? For instance, can the closure operation on a group be assigned a probability that says "how much" a member belongs to the group, kind of generalizing the rigid definition of making a member belong to the group. It will be great if experts here can point to such structures and recommend some books. Thanks.

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    $\begingroup$ It's not quite the same as what you suggest in your question, but I wonder if the notion of a hypergroup might be in the right spirit. You can think of a hypergroup as a weaker version of a group, in the sense that given two point masses $\delta_x$ and $\delta_y$ we no longer have a single "location" for the product (i.e. $\delta_{xy})$ but instead the "product" of $\delta_x$ and $\delta_y$ is a probability measure. $\endgroup$
    – Yemon Choi
    Jan 11, 2016 at 19:05
  • $\begingroup$ Thanks! I think hypergroup comes close. As I read and understand, hypergroup associates subsets to the product. I would like to read this definition and it will be great if you can point to some resources. $\endgroup$
    – Kasthuri
    Jan 12, 2016 at 12:09
  • $\begingroup$ I'm afraid I don't know much about the literature. There is a book by Bloom and Heyer which treats important parts of the general theory and also has examples, but I don't know if it is suitable as an introduction. $\endgroup$
    – Yemon Choi
    Jan 12, 2016 at 17:48

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