By Donald L. Cohn
Meant as a simple creation to degree idea, this textbook emphasizes these themes suitable and essential to the learn of study and likelihood concept. the 1st 5 chapters care for summary degree and integration. on the finish of those chapters, the reader will savour the weather of integration. bankruptcy 6, on differentiation, contains a remedy of adjustments of variables in Rd. a different function of the ebook is the introductory, but complete remedy of integration on in the neighborhood Hausdorff areas, of the analytic and Borel subsets of Polish areas, and of Haar measures on in the neighborhood compact teams. degree conception offers the reader with instruments wanted for learn in different components of present curiosity, particularly harmonic research and chance conception, and is a beneficial reference instrument.
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Extra info for Measure Theory
Example text
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.