By Geiss C., Geiss S.

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The convexity of x → |x|p gives that a+b 2 p ≤ |a|p + |b|p 2 and (a+b)p ≤ 2p−1 (ap +bp ) for a, b ≥ 0. Consequently, |f +g|p ≤ (|f |+|g|)p ≤ 2p−1 (|f |p + |g|p ) and ❊|f + g|p ≤ 2p−1(❊|f |p + ❊|g|p). 6. SOME INEQUALITIES 61 Assuming now that (❊|f |p ) p + (❊|g|p ) p < ∞, otherwise there is nothing to prove, we get that ❊|f +g|p < ∞ as well by the above considerations. Taking 1 < q < ∞ with p1 + 1q = 1, we continue by 1 ❊|f + g|p = ≤ = ≤ 1 ❊|f + g||f + g|p−1 ❊(|f | + |g|)|f + g|p−1 ❊|f ||f + g|p−1 + ❊|g||f + g|p−1 (❊|f |p ) ❊|f + g|(p−1)q (❊|g|p ) ❊|f + g|(p−1)q 1 p 1 q 1 p 1 q , ¨ lder’s inequality.

For the above types of convergence the random variables have to be defined on the same probability space. There is a variant without this assumption. 2 [Convergence in distribution] Let (Ωn , Fn , Pn ) and (Ω, F, P) be probability spaces and let fn : Ωn → ❘ and f : Ω → ❘ be random variables. Then the sequence (fn )∞ n=1 converges in distribution d to f (fn → f ) if and only if ❊ψ(fn) → ψ(f ) as n → ∞ for all bounded and continuous functions ψ : ❘ → ❘. 63 64 CHAPTER 4. MODES OF CONVERGENCE We have the following relations between the above types of convergence.

32 CHAPTER 2. 1 [measurable map] Let (Ω, F) and (M, Σ) be measurable spaces. A map f : Ω → M is called (F, Σ)-measurable, provided that f −1 (B) = {ω ∈ Ω : f (ω) ∈ B} ∈ F for all B ∈ Σ. 2 Let (Ω, F) be a measurable space and f : Ω → ❘. Then the following assertions are equivalent: (1) The map f is a random variable. (2) The map f is (F, B(❘))-measurable. 3 Let (Ω, F) and (M, Σ) be measurable spaces and let f : Ω → M . Assume that Σ0 ⊆ Σ is a system of subsets such that σ(Σ0 ) = Σ. If f −1 (B) ∈ F for all B ∈ Σ0 , f −1 (B) ∈ F for all B ∈ Σ.

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

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