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MATH 507/420 - Assignment #5 Due on Friday November 25, 2011 Name —————————————– Student number ————————— 1 Problem 1: Let (X, M, µ) be a measure space. Prove that L∞ (X, µ) is complete. 2 Problem 2: Let µ) be a measure space and p ∈ (0, 1). Prove that ||f ||p = R (X,p M,1/p ( X |f | dµ) does not define a norm on Lp (X, µ). 3 Problem 3: Prove the uniqueness assertion in the Lebesgue decomposition theorem. 4 Problem 4: For p ∈ (1, ∞), show that if the absolutely continuous function f on [a, b] is the indefinite integral of an Lp ([a, b]) function, then there is a constant K such that for any partition {x0 , x1 , ..., xn } of [a, b], n X |f (xk ) − f (xk−1 )|p ≤ K. p−1 |x k − xk−1 | k=1 5 Problem 5: Let (X, M) be a measurable space and µ and ν two σ-finite measures on it. If µ ν and ν µ, show that dν dµ · = 1 a.e. w.r.t. µ. dµ dν 6 Problem 6: Formulate and prove an extension of Hölder’s inequality for the product of three functions.