Document Type : Research Paper

**Authors**

Vali-e-Asr University of Rafsanjan, Department of Mathematics, P. O. Box 7713936417, Rafsanjan, Iran.

**Abstract**

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.

**Keywords**

**Main Subjects**

[1] G.D. Birkhoff, *Proof of the ergodic theorem,* Proc. Natl. Acad. Sci. USA., 17 (1931), pp. 656-660.

[2] R. Bracewell, *The Fourier Transform and Its Applications*, McGraw-Hill, (1986).

[3] P. Dirac, *The Principles of Quantum Mechanics*, Oxford at the Clarendon Press, (1958).

[4] I.M. Gelfand and G. Shilov, *Generalized Functions*, Academic Press, (1966-1968).

[5] E. Hopf, *On the time average in dynamics*, Proc. Nati. Acad. Sci. USA., 18 (1932), pp. 93-100.

[6] G.W. Mackey, *Ergodic theory and its signiificance for statistical mechanics and probability theory*, Adv. Math., 12 (1974), pp. 178-268.

[7] J.C. Maxwell, *On Boltzmann's theorem on the average distribution of energy in a system of material points*, Trans. Camb. Phil. Soc., 12 (1879), pp. 547-575.

[8] K. Peter, *Lectures on Ergodic Theory*, Springer, (1997).

[9] O. Sarig, *Lecture Notes on Ergodic Theory*, Springer, (2008).

[10] L. Schwartz, *Theorie Des Distributions*, Hermann, (1950).

[11] Y.G. Sinai, *Dynamical systems II. Ergodic theory with applications to dynamical systems and statistical mechnics*, Springer, (1989).

[12] A.M. Vershik, *Asymptotic Combinatorics with Applications to Mathematical Physics*, Springer, (2003).

[13] J.W. Von Neumann, *Proof of the quasi-ergodic hypothesis*, Proc. Nati. Acad. Sci. USA., 18 (1932), pp. 70-82.

[14] P. Walters, *An Introduction to Ergodic Theory*, Springer, (1982).

[15] E. Zeidler, *Quantum Field Theory: Basic in Mathematics and Physics*, Springer, (2007).