20 06 2024 E TS

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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Exercises Prof. F. Punzo, G. Verzini, June 20, 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 set function µ : P(N0) → [0, +∞), defined by µ(E) = X n∈E, n odd 5 n2 ∀E ∈ P (N0), where P(N0) denotes the power set of N0 = N \ {0}. Denote by δ3 the Dirac mass concentrated at 3. 1. Prove that µ is a measure on the measurable space ( N0, P(N0)). 2. Does there exist dδ3 dµ ? Justify carefully your answer and, if possible, compute it. Solution. (1) µ is a positive measure defined on P(N0), indeed µ(∅) =P n∈∅ 5 n2 = 0 and µ is σ-additive: take {Ei}i∈N ⊂ P(N0) disjoint family of measurable sets, then µ [ i∈N Ei ! = X n∈S i Ei, n odd 5 n2 = X i∈N X n∈Ei, n odd 5 n2 = X i∈N µ(Ei), where the validity of the second equality follows by the disjoint character of the family {Ei}i. (2) We check if the assumptions of the Radon-Nikodym theorem are satisfied, so to get the existence (and uniqueness a.e.) of the Radon-Nikodym derivative dδ3 dµ . Is µ σ -finite? Yes, actually µ is a finite measure, indeed for any E ∈ P (N0) we have µ(E) = X n∈E, n odd 5 n2 ≤ +∞X n=1 5 n2 , which is finite (being a convergent generalized harmonic series). Is δ3 ≪ µ? We notice that µ(E) = 0 if and only if E ∩ {n ∈ N0 : n odd} = ∅ (indeed 5 n2 > 0 for any n ∈ N0). In particular µ(E) = 0 ⇒ E ∩ {3} = ∅ ⇒ δ3(E) = 0 since δ3(E) = 1 if 3 ∈ E, while δ3(E) = 0 if 3 ̸∈ E. Hence, we can apply the Radon-Nikodym…