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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Exercises Prof. F. Punzo, G. Verzini, September 9, 2024 E1 E2 E3 Surname/Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Id. No. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Solutions must be written ONL Y on these sheets, under the exercise and in the back.] [Solutions can be written in English or in Italian.] Exercise 1. [5 points] Consider the measurable space ( Z, P(Z)), where P(Z) denotes the power set of the integer number set Z. Let µ : P(Z) → [0, +∞] be the counting measure and δ2 be the Dirac delta measure concentrated at 2. 1. Is µ ≪ δ2? Justify your answer. 2. Is δ2 ≪ µ? Justify your answer. 3. Is µ σ -finite? Is δ2 σ-finite? Justify your answers. 4. What can you conclude about the existence of dµ dδ2 and of dδ2 dµ ? Justify carefully your answer and, if possible, compute it. Solution. (1) We notice that, taking for instance the set E = {3}, then δ2(E) = 0 since 2 ̸∈ E, however µ(E) = 1 ̸= 0. Thus µ cannot be absolutely continuous w.r.t. δ2. (2) We notice that we have the following chain of implications µ(E) = 0 ⇒ E = ∅ ⇒ δ2(E) = 0. Thus δ2 is absolutely continuous w.r.t. µ. (3) δ2 is a finite measure, indeed δ2(Z) = 1 < +∞, in particular it is σ-finite. The measure µ is σ-finite, indeed we can write for instance Z = ∪n∈NEn, with En := {−n, n}, where En ∈ P (Z) and µ(En) = 2 < +∞ for any n. (4) By definition of Radon-Nikodym derivative, we can say that dµ dδ2 cannot exists, since if it exists then we should have µ ≪ δ2 which is false. However, since δ2 ≪ µ and µ is σ-finite, we can apply the Radon-Nikodym theorem to affirm that there exists dδ2 dµ , i.e. a measurable map from Z to [0, +∞] such that…