By Seidenberg A.

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Thus, Lp (X, Σ, μ) becomes a vector space under the usual function addition and scalar multiplication for any p ∈ [1, ∞]. We shall sometimes write Lp instead of Lp (X, Σ, μ) if the measure space is understood, Lp (X) if Σ and μ are understood, Lp (μ) if X and Σ are understood, or Lp (Σ) if X and μ are understood. We note that L1 is exactly the space of all μ-integrable functions. The number f p |f |p dμ = 1/p is called the Lp -norm of f ∈ Lp for X p < ∞ and the number g ∞ = ess sup |g(x)| is referred to as the L∞ -norm of x∈X g ∈ L∞ .

Given a 48 Chapter 3 Rudiments of Ergodic Theory continuous transformation S : X → X, we denote by U ≡ US : C(X) → C(X) the composition operator deﬁned by U f (x) = f (S(x)). In the next chapter, when we study Frobenius-Perron operators, this operator U will be called the Koopman operator which is deﬁned on L∞ (X) with a given measure μ on X. , U (f · g) = (U f )(U g). It can also be used in the statement of the ergodic theorems in the previous section. 1 Let Y = {0, 1, · · · , k − 1} with the discrete topology.

2 By the same token, any translation mapping S(x) = ax on a compact group G preserves the Harr measure on G, where a is a ﬁxed element of G. 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.

### A new decision method for elementary algebra by Seidenberg A.

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