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We develop a new duality between endomorphisms of measure spaces, on the onehand, and a certain family of positive operators, called transfer operators,acting in spaces of measurable functions on, on the other. A framework ofstandard Borel spaces is adopted; and this generality is wide enough to cover ahost of applications. While the mathematical structures of positive operators,endomorphisms, transfer operators, measurable partitions, and Markov processesarise in a host of settings, both pure and applied, we propose here a unifiedstudy. This is the general setting of dynamics in Borel measure spaces. Hencethe corresponding linear structures are infinite-dimensional. Nonetheless, weprove a number of analogues of the more familiar finite-dimensional settings,for example, the Perron-Frobenius theorem for positive matrices, and thecorresponding Markov chains. Tools from the theory of operators in Hilbertspace of special significance to us will be the use of a certain universalHilbert space, as well as classes of operators in it, directly related to thecentral theme of duality for transfer operators. From ergodic theory, weaddress such questions as measurable cross sections, partitions, and Rohlinanalysis of endomorphisms of measure spaces. While there are classical theoremsdealing with analogous questions for automorphisms of measure spaces, asystematic study of endomorphisms is of more recent vintage;-- in its infancy.In order to make the exposition accessible to students and to researchers inneighboring areas, we have included a number of explicit examples andapplications.

Infinite-dimensional transfer operators, endomorphisms, and measurable partitions