MATH 525a SAMPLE FINAL EXAM Fall 2013 Prof. Alexander NOTES: (i) This exam is from several years ago. This semester problem (5) was given as homework, and (4) is similar to Ch. 3 #12. (ii) The final is in the lecture room, Monday December 16, 11 am–1 pm. It is open book, meaning you can use the Folland text, your lecture notes, homework and solutions. You cannot use other published materials (books, material from the internet, etc.) The final covers the full semester, but with more emphasis on the second half (after the midterm.) (iii) It’s not expected that one could do all 6 problems in 2 hours, but it should be feasible to do at least 4. (1) Let X be a set and A ⊂ X. Define the collection of sets which contain either “all or none” of A: M = {E ⊂ X : A ⊂ E or E ∩ A = φ}. (a) Show that M is a σ-algebra. (b) Which functions f : X → R are measurable on (X, M)? Give a description that is specific to this problem, not just the general definition of measurable function. (2) Let fn , f, g be real-valued measurable functions on (X, M, µ). Suppose fn → f in measure and |fn | ≤ g a.e. for all n. (a) Show that |f | ≤ g.R R (b) Assume also that g 2 dµ < ∞. Show that (fn − f )2 dµ → 0. (3) Let F1 , F2 , ... and F all be in NBV, and suppose Fk → F pointwise. Show that TF (x) ≤ lim inf TFk (x) for all x. k→∞ (See page 102, 103 for the definitions of TF and NBV.) HINT: Show that for arbitrary > 0, TF (x) − ≤ lim inf k TFk (x). (4) Suppose µ1 , ν1 are positive finite measures on (X1 , M1 ) and µ2 , ν2 are positive finite measures on (X2 , M2 ), with ν1 µ1 and ν2 µ2 . Show that ν1 × ν2 µ1 × µ2 . HINT: It’s enough to show that for some f , Z ν1 × ν2 (E) = f d(µ1 × µ2 ) E for all E ∈ M1 × M2 . See if you can guess what f should be, then prove you’re right. 1 (5) Let us say that two signed measures ν1 , ν2 on (X, M) are compatible if there exists a decomposition X = P ∪ N which is a Hahn decomposition for both ν1 and ν2 . According to Proposition 3.14 page 94, for ν1 , ν2 finite signed measures, (∗) |ν1 + ν2 |(E) ≤ |ν1 |(E) + |ν2 |(E) for all E. dν Let µ = |ν1 | + |ν2 | and let fj = dµj , j = 1, 2. We write [fj > 0] as a shorthand for {x ∈ X : fj (x) > 0}, and similarly for [fj < 0]. Show that the following are equivalent: (i) ν1 and ν2 are compatible; (ii) equality holds in (*) for all E; (iii) µ([f1 > 0] ∩ [f2 < 0]) = µ([f1 < 0] ∩ [f2 > 0]) = 0. HINT: Show (i) ⇔ (iii) and (ii) ⇔ (iii). To show (ii) implies (iii), show “not (iii)” implies “not (ii).” (6) Let (X, M, µ) be a measure space, with µ finite, and let ϕ be a bounded linear functional on L1 (µ). Define ν on M by ν(E) = ϕ(χE ). (a) Show that ν is a finite signed measure. HINT: Use Prop. 5.2. (b) Show that ν µ. R dν dν dν is a bounded function. HINT: dµ ≤ M a.e. if and only if E dµ dµ ≤ (c) Show that dµ R M dµ for all E. (Why? Explain, if you this hint.) E R use dν (d) Show that ϕ is given by ϕ(f ) = f dµ dµ for all f ∈ L1 (µ). HINT: Consider f = χE first. 2