some observations on dirac measure-preserving transformations and their results

Dirac measure is an important measure in many related branches to mathematics. the current paper characterizes measure-preserving transformations between two dirac measure spaces or a dirac measure space and a probability measure space. also, it studies isomorphic dirac measure spaces, equivalence dirac measure algebras, and conjugate of dirac measure spaces. the equivalence classes of a dirac measure space and its measure algebras are presented. then all of measure spaces that are isomorphic with a dirac measure space are characterized and the concept of a dirac measure class is introduced and its elements are characterized. more precisely, it is shown that every absolutely continuous measure with respect to a dirac measure belongs to the dirac measure class. finally, the relation between dirac measure preserving transformations and strong-mixing is studied.