From: "Noel Vaillant" Subject: Re: Lebesgue integral Date: Wed, 26 Apr 2000 00:28:00 +0100 Newsgroups: sci.math Summary: [missing] > Let and be a sequence of integrable functions with f_n->f > and g_n->g (also integrable functions), |f_n|<=g_n and int(g)=lim > int(g_n) (where int(...) means integral of ....). Then int(f)=lim > int(f_n). > > Any hints? > Personally, I would revisit the proof of the dominated convergence theorem as follows [I think it works]: Define h_n=g_n+g-|f_n-f|. Then h_n >= 0 for all n, and we can apply Fatou, lemma to the sequence . We obtain: int(2g)<=liminf (int(g)+int(g_n)-int(|f_n-f|)) = int(g) + liminf(int(g_n)-int(|f_n-f|)) (*) Using the fact that int(g_n) converges to the real number int(g), the inequality can be re-expressed as: 2int(g)<=2int(g)+liminf(-int(|f_n-f|)) And we obtain limsup(int(|f_n-f|))=0 In particular int(f_n)->int(f). I hope this works. Regards. Noel. -- Noel Vaillant http://www.probability.net