From: Robin Chapman <rjc@maths.ex.ac.uk>
Newsgroups: sci.math
Subject: Re: Fatou's Lemma...proof in Royden
Date: Mon, 30 Nov 1998 09:22:52 GMT

In article <3661cba3.231605525@news.wenet.net>,
  cking@wenet.net (Sheila King) wrote:
> I'm doing my homework for my Real Analysis course, and am studying my
> notes and the text to help me think of ways to attack the assigned
> proofs. We are using Royden, 3rd ed, publ. Prentice-Hall
>
> On pp. 86-87 is a proof of Fatou's Lemma, and somehow I am not
> understanding all of the nuances of the proof. I'm hoping someone here
> will be able to help clarify my muddled thinking.
>
> Fatou's Lemma asserts: f_n is a seq. of non-neg measurable functions.
> f_n --> f almost everywhere on a set E. Then
> Integral of f on E <= lim inf of Integral of f_n on E.
>
> The method of the proof is as follows (somewhat sketchy):
> (ignore set of meas. zero where f_n do not converge to f since on that
> set there is no contribution to the integral)
>
> Let h be a bounded function  not greater than f
> Construct a sequence h_n(x) = min{h(x), f_n(x)}
> Then the h_n are bounded by the bound for h and h_n-->h
>
> By the Bounded Convergence Theorem we have
> Integral of h on E = lim(n-->infinity) of Integral of h_n on E
>
> All of the above I understand.
>
> There is an additional statement, however, tacked on to the last
> statement about the integrals of h which I have not been able to figure
> out how it is justified, despite thinking about it for a couple of
> hours.
>
> Royden says that
> lim(n-->infinty)of Integral of h_n <= lim inf of Integr. of f_n
>
> He then goes on to take supremum and finishes the proof in the next
> step.

Let's look at Royden's las two sentences:

(i) integral_E h = integral_{E'} h : this is true as h(x) vanishes for x in E
but not in E'.

(ii) integral_{E'} h = lim integral_{E'} h_n : this is true by Prop. 6
as h_n(x) -> h(x) for all x in E' and |h_n(x)| <= |h(x)| <= M for some M.

(iii) lim integral_{E'} h_n <= lim inf integral_E f_n : this
follows since h_n <= f_n. In the definition of integral f_n (p.85)
integral h_n is one of the set of numbers that integral f_n is a sup of.
Hence integral_{E'} h_n <= integral_{E'} f_n . Also integral_{E'} f_n
= integral_E f_n. Hence lim inf integral_{E'} h_n <= lim inf integral_{E'} f_n
(but the first of these lim infs is actually a lim).

(iv) the last sentence: integral_E f is the supremum of all the integral_E h.
Thus it is <= the lim inf as all the integral_E h are.

Robin Chapman                           + "They did not have proper
SCHOOL OF MATHEMATICal Sciences         -  palms at home in Exeter."
University of Exeter, EX4 4QE, UK       +
rjc@maths.exeter.ac.uk                  -  Peter Carey,
http://www.maths.ex.ac.uk/~rjc/rjc.html +    Oscar and Lucinda, chapter 20

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