measure-preserving transformation

The term endomorphism derives from the Greek adverb endon ("inside") and morphosis ("to form" or "to shape"). In algebra, an endomorphism of a group, module, ring, vector space, etc. is a homomorphism from one object to itself (with surjectivity not required). In ergodic theory, let X be a set, F a sigma-algebra on X and m a probability measure. A map T:X->X is called an endomorphism (or measure-preserving transformation) if 1.T is surjective, 2.T

endomorphism ring | measurable function | morphism | sigma-algebra | surjection