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Departments of Mathematics Fall 2015 Montana State University Prof. Kevin Wildrick Measure Theory Problem Set 11 Due Friday, November 20 th , 11:00 am. The symbol (?) indicates that this problem must be solved and turned in. Other problems should be solved but need not be turned in. 1. Let λ be charge on a set X equipped with a σ-algebra Σ. Show that λ+ and λ− are independent of the choice of the Hahn decomposition used to define them. 2. (?) Exercises 8A and 8B in Bartle. 3. (?) Let (X, Σ, µ) be a measure space, and let f ∈ L1 (X, Σ, µ). Define µf : Σ → R by Z µf (E) := f dµ. E • Show that µf is a charge. • Do the sets P = {x ∈ X : f (x) ≥ 0} and N = {x ∈ X : f (x) < 0} define a Hahn decomposition for µf ? • Show that for any E ∈ Σ, µ+ f (E) Z f = E + dµ, and µ− f (E) Z = E f − dµ.