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. Origins of ergodic theory. Our results will be con­ cerned with 1-1 invertible transformations, 7\ of a measure space X, of total measure 1, where T and T*1 transform each measurable set onto a measurable set of the same measure. We will be concerned with the behavior of T only up to sets of measure 0. That is, we will say that Tx acting on Xi is isomorphic to r2 acting on X 2 if we can find XiC^i and X2QX2 of measure 1 and invariant under T\ and T2t respectively, and if there is an invertible, measure-preserving transformatio n \p of X\ onto X2 such that for all x in Xi we have \f/Ti(x) = T$p{x).

Some new results in the Kolmogorov-Sinai theory of entropy and ergodic theory