1. The problem statement, all variables and given/known data This comes courtesy of Royden, problem 4.14. a.Show that under the hypothesis of theorem 17 we have [tex]\int |fn-f| \rightarrow 0[/tex] b.Let <fn> be a sequence of integrable functions such that [tex]fn \rightarrow f a.e.[/tex] with f integrable. Then [tex]\int |fn-f| \rightarrow 0[/tex] if and only if [tex] \int |fn| \rightarrow \int |f|[/tex] 2. Relevant equations Theorem 17: Leg <gn> be a sequence of integrable functions which converges to an integrable function g. Let <fn> be a sequence of measurable functions such that [tex] |fn|\leq gn [/tex] and <fn> converges to f a.e. If [tex]\int g = lim \int gn[/tex] Then [tex]\int f = lim \int fn[/tex] 3. The attempt at a solution My problem in part a is the same as the "if" in part b. The only if seems fairly trivial since |a-b|>=| |a| - |b| |. Unfortunately that is partially the problem. Every attempt I've made to go in the other direction reduces to that same inequality. My basic thought process far has been to start with [tex]\int |f| = lim \int |fn| [/tex] which we have by assumption, and then conclude that for some e>0 theres an N such that [tex]| \int |fN| - |f| |< e[/tex] Obviously this does not get me where I want to go. If I somehow had fn as an increasing function I coud keep going, but I see no real way to do this.