From: "Noel Vaillant" <vaillant@netcomuk.co.uk>
Subject: Re: Lebesgue integral
Date: Wed, 26 Apr 2000 00:28:00 +0100
Newsgroups: sci.math
Summary: [missing]

> Let <f_n> and <g_n> 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 <h_n>. 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