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By Eduard L. Stiefel

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Any μ ∈ M (X) is called a Borel probability measure on X. The following Riesz’s representation theorem [57] basically gives an isomorphic relation between M (X) and the collection of all normalized positive linear functionals on C(X). 6 (Riesz’s representation theorem) Let X be a compact metric space and let w be a continuous linear functional on C(X) such that w(f ) 0 for f 0 and w(1) = 1. Then, there exists a unique μ ∈ M (X) such f dμ for all f ∈ C(X). 1 If μ and ν are two Borel probability measures on X such that f dν for all f ∈ C(X), then μ = ν.

4 A sequence of functions fn ∈ Lp , 1 convergent to f ∈ Lp if p < ∞ is weakly lim fn , g = f, g , ∀ g ∈ Lp , n→∞ and is strongly convergent to f ∈ Lp if lim n→∞ fn − f p = 0. 3), we see that strong convergence implies weak convergence. The converse is not true in general, as demonstrated by the classic example of fn (x) = sin(nx) in L2 (0, 1) [109]. 3 Functions of Bounded Variation in One Variable The concept of variation plays an important role in the compactness argument for L1 spaces, and thus will be a key concept for studying Frobenius-Perron operators that are defined on L1 spaces.

More generally, any continuous isomorphism of G is Harr measure invariant. Our last example concerns the logistic model that we encountered in Chapter 1. 6 The quadratic mapping S(x) = 4x(1 − x), ∀ x ∈ [0, 1] preserves the absolutely continuous probability measure μ of the interval [0, 1] given by 1 dx, ∀ A ∈ B. 2 Ergodicity, Mixing and Exactness 39 Proof. 1) is tedious even for A = [a, b]. A direct computation shows that 1 1 g(x)dμ(x) = 0 g(x) 0 π 1 2 dx = π x(1 − x) π 2 g(sin2 θ)dθ 0 and 1 1 g(S(x))dμ(x) = 0 0 = 2 π g(4x(1 − x)) π 2 0 π 1 dx x(1 − x) g(sin2 (2θ))dθ = 1 π π g(sin2 t)dt = 0 so the result follows.

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