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Submit 12 months notice: First released in 2006
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The second one follows from the definition of lim sup and lim inf, the third one can be proved like the first one. , |Zk | ≤ g and | lim inf fn | ≤ g. 5 gives that ❊ limninf fn = lim ❊Zk = lim ❊ n≥k inf fn k k ≤ lim inf k n≥k ❊fn = lim inf ❊fn . 7 [Lebesgue’s Theorem, dominated convergence] Let (Ω, F, P) be a probability space and g, f, f1 , f2 , ... s. s. Then f is integrable and one has that ❊f = lim ❊fn. n Proof. Applying Fatou’s Lemma gives ❊f = ❊ lim inf fn n→∞ ≤ lim inf ❊fn ≤ lim sup ❊fn ≤ n→∞ n→∞ ❊ lim sup fn = ❊f.

3. Now we state some general properties of measurable maps. 5 Let (Ω1 , F1 ), (Ω2 , F2 ), (Ω3 , F3 ) be measurable spaces. Assume that f : Ω1 → Ω2 is (F1 , F2 )-measurable and that g : Ω2 → Ω3 is (F2 , F3 )-measurable. Then the following is satisfied: (1) g ◦ f : Ω1 → Ω3 defined by (g ◦ f )(ω1 ) := g(f (ω1 )) is (F1 , F3 )-measurable. (2) Assume that P is a probability measure on F1 and define µ(B2 ) := P ({ω1 ∈ Ω1 : f (ω1 ) ∈ B2 }) . Then µ is a probability measure on F2 . The proof is an exercise.

But now we get E gn (η)dPϕ (η) = Pϕ (B) = P(ϕ−1 (B)) = = E E 1Iϕ−1 (B) (η)dP(η) 1IB (ϕ(η))dP(η) = E gn (ϕ(η))dP(η). Let us give two examples for the change of variable formula. 2 [Computation of moments] We want to compute certain moments. Let (Ω, F, P) be a probability space and ϕ : Ω → ❘ be a random variable. 5. FUBINI’S THEOREM 51 for all ∞ < a < b < ∞ where p : ❘ → [0, ∞) is a continuous function such ∞ that −∞ p(x)dx = 1 using the Riemann-integral. 2. 1 and that Pϕ = ∞ pk δηk k=1 with pk ≥ 0, ∞ k=1 pk = 1, and some ηk ∈ E (that means that the image measure of P with respect to ϕ is ’discrete’).

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