From: Chris Hillman <hillman@math.washington.edu>
Subject: Re: Questions about Lesbegue measure
Date: Sat, 1 Apr 2000 18:10:50 -0800
Newsgroups: sci.math
Summary: [missing]

On Sun, 2 Apr 2000, *Paragon* wrote:

> First off, are there any sites with info, theorems, etc. about Lesbegue
> measure?

Hmm... I can list some sites dealing with undergraduate real analysis:

 http://www.shu.edu/html/teaching/math/reals/gloss/index.html

 http://aurora.phys.utk.edu/~forrest/papers/fourier/index.html

(wasn't someone asking about the Shannon Sampling Theorem?)

 http://psych.hanover.edu/Krantz/fourier/square.html

 http://www.amara.com/IEEEwave/IEEEwavelet.html
 
> Secondly, I'm having trouble proving a theorem dealing with measure on
> the real number line:

Is this homework?

If so, I'll recommend that you get Folland, Real Analysis, Second Edition.  
I and a number of others who have studied the most popular graduate level
textbooks agree that while all the good books offer something valuable and
unique, this book is overall the best one around.  (I was discussing this
with someone just the other day, in fact--- mathematicians like to talk
about their favorite math books, like other people like to chat about why
they like the music of their favorite pop groups.)
 
Chris Hillman

Home Page: http://www.math.washington.edu/~hillman/personal.html
==============================================================================

From: chuleta2099@my-deja.com
Subject: Re: Questions about Lesbegue measure
Date: Mon, 03 Apr 2000 02:08:29 GMT
Newsgroups: sci.math

In article <8c66cl$pfq$1@nnrp1.deja.com>,
*Paragon* <karkadann@mailcity.com> wrote:
> First off, are there any sites with info, theorems, etc. about Lesbegue
> measure?

I am putting my class notes for real analysis on the web:

http://www.math.uiuc.edu/~uavalos/class/441/

You will probably find these more useful than most stuff on the web
since alot of the major theorems are proven in painstaking detail.  You
need to be able to view dvi files or convert dvi files to postscript
format.

> Secondly, I'm having trouble proving a theorem dealing with measure on
> the real number line:
>
> If M is a bounded measureable set and each of f and g is a bounded
> measureable function with domain M, then
> 1.) f+g is a measurable function
> 2.) integral f + integral g = integral (f+g),
> where integral is over all of M.
>
> It's rather complicated to type here, but the definition of Lesbegue
> measure given in class is different than the standard definition given
> in analysis books, but it be similar in some regards.
>
> Could someone give a brief outline of how to prove this?
>

It depends on what your definition of "integral f" is---usually you are
approximating by simple or step functions.  For example
integral f=sup{integral g:g\le f & g simple}.  To prove (2), you "play"
a sup or inf game---see royden for details (my notes, for some wierd
reason, take a more complicated approach).


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