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This equality is true for any Riemann integrable function I since any Riemann integrable function is automatically Lebesgue integrable. An analogous connection exists in higher dimensions. t(dx), 24 2. The Toolbox is a finite measure. In fact, by the definition of the Lebesgue integral it is clear that I'J(A) is nonnegative and finite, and from property (15) it is also additive. e. 2, and I'J(A) = 0 whenever J'(A) = 0. This observation that every integrable nonnegative function defines a finite measure can be reversed by the following theorem, which is of fundamental importance for the development of the Frobenius-Perron operator.

However, as a preliminary step, we develop the more general concept of the Markov operator and derive some of its properties. Our reasons for this approach are twofold: First, as will become clear, many concepts concerning the asymptotic behavior of densities may be equally well formulated for both deterministic and stochastic systems. Second, many of the results that we develop in later chapters concerning the behavior of densities evolving under the influence of deterministic systems are simply special cases of more general results for stochastic systems.

0 Markov operators have a number of properties that we will have occasion to use. ::. ::. g(x). 2) is said to be monotonic. To show the monotonicity of P is trivial, since (! ::. ::. 0. To demonstrate further inequalities that Markov operators satisfy, we offer the following proposition. 1. 5) and (M4) IlP/II::;; 11/11. 6) Proof. These inequalities are straightforward to derive. 4) is obtained in an analogous fashion. 5) follows from (Ml) and (M2), namely, IP/1 = (Pj)+ +(PI)-::;; Pj+ + PJ= P(J+ + /-) = PI/I.