Is there a precise definition for the statement that two differently worded probability problems are "equivalent"? One technique of (purportedly) solving a controversial probability problem is to propose an "equivalent" problem whose solution is not controversial. (e.g. The Sleeping Beauty thread: page 22 post #422https://www.physicsforums.com/threads/the-sleeping-beauty-problem-any-halfers-here.916459/page-22 ). However, my impression of this technique is that people just assert they have created an equivalent problem and hope the reader will say Ah-ha!. If the specifics of two probability problems are given in the form of probability spaces, can we define their "equivalence" by the existence of some sort of mapping between the two spaces ? Can we define the concept of "homomorphic" and "isomorphic" for probability problems?