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By Geiss C., Geiss S.

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Proof. We simply have λP({ω : f (ω) ≥ λ}) = λ❊1I{f ≥λ} ≤ ❊f 1I{f ≥λ} ≤ ❊f. 2 [convexity] A function g : ❘ → ❘ is convex if and only if g(px + (1 − p)y) ≤ pg(x) + (1 − p)g(y) for all 0 ≤ p ≤ 1 and all x, y ∈ ❘. Every convex function g : ❘ → ❘ is (B(❘), B(❘))-measurable. 3 [Jensen’s inequality] If g : f : Ω → ❘ a random variable with ❊|f | < ∞, then ❘ → ❘ is convex and g(❊f ) ≤ ❊g(f ) where the expected value on the right-hand side might be infinity. 6. SOME INEQUALITIES 59 Proof. Let x0 = ❊f . Since g is convex we find a “supporting line”, that means a, b ∈ ❘ such that ax0 + b = g(x0 ) and ax + b ≤ g(x) for all x ∈ ❘.

2 [convexity] A function g : ❘ → ❘ is convex if and only if g(px + (1 − p)y) ≤ pg(x) + (1 − p)g(y) for all 0 ≤ p ≤ 1 and all x, y ∈ ❘. Every convex function g : ❘ → ❘ is (B(❘), B(❘))-measurable. 3 [Jensen’s inequality] If g : f : Ω → ❘ a random variable with ❊|f | < ∞, then ❘ → ❘ is convex and g(❊f ) ≤ ❊g(f ) where the expected value on the right-hand side might be infinity. 6. SOME INEQUALITIES 59 Proof. Let x0 = ❊f . Since g is convex we find a “supporting line”, that means a, b ∈ ❘ such that ax0 + b = g(x0 ) and ax + b ≤ g(x) for all x ∈ ❘.

Then we define the projections Πn : ❘◆ → ❘ given by Πn (x) := xn , that means Πn filters out the n-th coordinate. , B ∈ B(❘) , 38 CHAPTER 2. 5. Finally, let P1 , P2 , ... 14) we B(❘). Using Carathe find an unique probability measure P on B(❘◆ ) such that P(B1 × B2 × · · · × Bn × ❘ × ❘ · · · ) = P1(B1) · · · Pn(Bn) for all n = 1, 2, ... , xn ∈ Bn . 8 [Realization of independent random variables] Let (❘◆ , B(❘◆ ), P) and πn : Ω → ❘ be defined as above. Then (Πn )∞ n=1 is a sequence of independent random variables such that the law of Πn is Pn , that means P(πn ∈ B) = Pn(B) for all B ∈ B(❘).

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An introduction to probability theory by Geiss C., Geiss S.


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