homework7 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Problem Set 7 due 10/29/2008 Readings: Notes for lectures 11-13 (you may skip the proofs in the notes for lecture 11). Optional additional readings: Adams & Guillemin, Sections 2.2-2.3, skim Section 2.5.± For a full development of this material, see [W], Sections 5.1-5.9, 6.0-6.3, 6.5,± 6.12, 8.0-8.4. ± Exercise 1. Show that if g : Ω [0 , ] satisfies g dµ < , then g < , a.e. (i.e., the set { ω | g ( ω ) = ∞} has zero measure). Exercise 2. Let , F , P ) be a probability space. Let g : Ω R be a nonneg- ative measurable function. Let λ be the Lebesgue measure. Let f be a nonneg- ative measurable function on the real line such that f = 1 . For any Borel set A , let P 1 ( A ) = A f . Prove that P 1 is a probability measure. Exercise 3. (Impulses and Impulse Trains) Consider the real line, endowed with the Borel σ -field. For any c R , we define the Dirac measure (“unit impulse”) at c , denoted by δ c , to be the probability measure that satisfies δ c ( c ) = 1 . If we “place a Dirac measure” at each integer, n =1 n =1 we are led to the measure µ = , that is, µ ( A ) = ( A ) , for every
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