Groups with probability measures

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.

• 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. Jan 11, 2016 at 19:05
• 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. Jan 12, 2016 at 12:09
• 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. Jan 12, 2016 at 17:48