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27 (Monotone Class Theorem) Suppose H is a vector space of bounded functions Ω → R with 1 ∈ H and such that if (fn )n are non-negative elements of H such that f = supn fn is bounded, then f ∈ H. If H contains all indicators of sets in a π-system C, then H contains all bounded σ(C)-measurable functions. 4 Product measures 43 Proof The collection D = {A ⊂ Ω : 1A ∈ H} is a d-system (because of the properties of H) and contains C, hence also contains σ(C). Given a σ(C)-measurable function f with 0 ≤ f (ω) ≤ K for all ω ∈ Ω and some integer K, approximate f from below by simple functions K2n i i for each n.

Now use the above inequality p ||X|| Ω A p p with W = ||X||p Z1A ∈ Lq (Ω, F, Q) 1/q X p ZdP = (XY )dP = Ω Ω A = Ω pq ||Xp ||p W dQ ≤ q Y Xp W q dQ Ω p X 1A p dP ||X||p 1/q 1/q = ||X||p Ω Y q 1A dP ≤ ||X||p ||Y ||q . A familiar special case occurs when p = q = 2. 2 (Schwarz inequality) When X, Y ∈ L2 (Ω), then XY ∈ L1 (Ω) and |E[XY ]| ≤ ||XY ||1 ≤ ||X||2 ||Y ||2 . 1 Lp as a Banach space 51 We can now prove the triangle inequality for the norm X → ||X||p on Lp whenever p ≥ 1. 3 (Minkowski’s inequality) If p ≥ 1 and X, Y ∈ Lp (Ω), then ||X + Y ||p ≤ ||X||p + ||Y ||p .

12 Suppose p > 1, p1 + 1q = 1, and X, Y are nonnegative random variables with Y ∈ Lp (Ω), and λP (X ≥ λ) ≤ Y dP {X≥λ} for all λ ≥ 0. Then X ∈ Lp and ||X||p ≤ q ||Y ||p . Proof Fix n > 0 and let Xn = X ∧ n. Then Xn is bounded, hence in Lp . If we have proved our inequality for Y and Xn , the MCT shows that it also holds for Y and X = limn↑∞ Xn . We can thus take X ∈ Lp without loss. z Since for any z ≥ 0, p {z≥x} xp−1 dx = p 0 xp−1 dx = z p , we have, integrating z = X(ω), over Ω and using Fubini X(ω) X p dP = Ω xp−1 dx P (dω) p Ω 0 ∞ =p 0 ∞ =p 0 Ω 1{X(ω)≥x} P (dω) xp−1 dx xp−1 P (X ≥ x)dx.