Let (X, A,p) be a complete measure space and let f and g be nonnegative A-measurable functions on X such that fg e l'(X). For any > 0 let Ex = {r X g(x) > 1}. Show that the function F(X) = Sex f(x)dp() is finite for any 1 > 0 and $ 5(2)9(a)dp(x) = San F(A)dm (1). [0,00) Hint: Consider the set in Xx[0,00) that is "below the graph of g" as used in the previous homework. Integrate the pullback off on this set and apply the Tonelli theorem. Let (X, A,p) be a complete measure space and let f and g be nonnegative A-measurable functions on X such that fg e l'(X). For any > 0 let Ex = {r X g(x) > 1}. Show that the function F(X) = Sex f(x)dp() is finite for any 1 > 0 and $ 5(2)9(a)dp(x) = San F(A)dm (1). [0,00) Hint: Consider the set in Xx[0,00) that is "below the graph of g" as used in the previous homework. Integrate the pullback off on this set and apply the Tonelli theorem