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3 The Koopman Operator To close this chapter, we define a third type of operator closely related to the Frobenius-Perron operator. 1. 1) is called the Koopman operator with respect to S. 48 3. Markov and Frobenius-Perron Operators This operator wBB first introduced by Koopman [1931]. Due to the nonsingularity of S, U is well defined since It (x) = h(x) a. e. e. 4) so that U is adjoint to the Frobenius-Perron operator P. Property (K1) is trivial to check. e. e. 1), Uf(x) = f(S(x)). Finally, to obtain (K3) we first check it with g = 1A.

X) = oo even if fm f E £Pl n£P2. 2. 7. £) be a finite measure space and let Show that the function f E L 00 (X) be fixed. 1~p

If measure spaces (Xi, A, l'i), i = 1, ... 8), respectively, then there exists a unique extension of I' to a measure defined on A. 2, is called the product of the measure spaces (X11 A 1,JL1), ... I'd), or more briefly a product space. The measure I' is called the product measure. 8) it follows that JL(Xl X ••• X Xd) = JL(Xl) .. ·JL(Xd)· Thus, if all the measure spaces (Xi, A, l'i) are finite or probabilistic, then (X, A, JL) will also be finite or probabilistic. 2 allows us to define integration on the product space (X, A, JL) since it is also a measure space.

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