## Question

###### (20 points) Suppose that sequence of measurable functions fn Xx[0,&] is given such tat < fn+(z) < fn(r) for all _ eX Assume further that fi â‚¬ L'(X,e) i.e Jx f dp < &. Show thatlim +00J, dp Ks duHint: Use the convergence theorems! Bonus question (10 points): Show by counterex- ample that the conelusion is false if f 4 L'(Xp).

(20 points) Suppose that sequence of measurable functions fn Xx[0,&] is given such tat < fn+(z) < fn(r) for all _ eX Assume further that fi â‚¬ L'(X,e) i.e Jx f dp < &. Show that lim +00 J, dp Ks du Hint: Use the convergence theorems! Bonus question (10 points): Show by counterex- ample that the conelusion is false if f 4 L'(Xp).

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