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Question about Lebesque Integral
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PostPosted: Tue May 03, 2005 7:37 pm    Post subject: Question about Lebesque Integral Reply with quote
The Lebesque integral ,I, of function f with respect to measure u is
defined by

I (f du) = I (f+ du) - I (f- du)

where f+ and f- are the positive and negative parts of f, respectively.
Now my text (Bartle) states that "it is easy to see that if" f = v -
w, with v and w non-negative, then

I (f du)= I (v du) - I ( w du)

I don't see how this follows from the basic definition of I as supremum
of integrals of simple functions. Can someone please help with a proof?
Thanks.
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Igor Khavkine
science forum Guru


Joined: 01 May 2005
Posts: 607

PostPosted: Wed May 04, 2005 12:44 am    Post subject: Re: Question about Lebesque Integral Reply with quote
On 2005-05-03, agapito6314@aol.com <agapito6314@aol.com> wrote:
Quote:
The Lebesque integral ,I, of function f with respect to measure u is
defined by

I (f du) = I (f+ du) - I (f- du)

where f+ and f- are the positive and negative parts of f, respectively.
Now my text (Bartle) states that "it is easy to see that if" f = v -
w, with v and w non-negative, then

I (f du)= I (v du) - I ( w du)

I don't see how this follows from the basic definition of I as supremum
of integrals of simple functions. Can someone please help with a proof?

You can't. The definition using suprema of integrals of simple functions
works only if f itself is non-negative. To show the second equation,
split f into its positive and negative parts (where is v greater than w,
and vice versa?), them apply the definition for signed integrands (the
first equation).

Hope this helps.

Igor
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David C. Ullrich
science forum Guru


Joined: 28 Apr 2005
Posts: 2250

PostPosted: Wed May 04, 2005 1:23 pm    Post subject: Re: Question about Lebesque Integral Reply with quote
On 3 May 2005 14:37:47 -0700, agapito6314@aol.com wrote:

Quote:
The Lebesque integral ,I, of function f with respect to measure u is
defined by

I (f du) = I (f+ du) - I (f- du)

where f+ and f- are the positive and negative parts of f, respectively.
Now my text (Bartle) states that "it is easy to see that if" f = v -
w, with v and w non-negative, then

I (f du)= I (v du) - I ( w du)

I don't see how this follows from the basic definition of I as supremum
of integrals of simple functions. Can someone please help with a proof?

You got several replies the first time you posted this question.

Quote:
Thanks.


************************

David C. Ullrich
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Arturo Magidin
science forum Guru


Joined: 25 Mar 2005
Posts: 1838

PostPosted: Wed May 04, 2005 5:39 pm    Post subject: Re: Question about Lebesque Integral Reply with quote
In article <t5qh715ake49d5v9q9svd2rcrh2n35s7b9@4ax.com>,
David C. Ullrich <ullrich@math.okstate.edu> wrote:
Quote:
On 3 May 2005 14:37:47 -0700, agapito6314@aol.com wrote:

The Lebesque integral ,I, of function f with respect to measure u is
defined by

I (f du) = I (f+ du) - I (f- du)

where f+ and f- are the positive and negative parts of f, respectively.
Now my text (Bartle) states that "it is easy to see that if" f = v -
w, with v and w non-negative, then

I (f du)= I (v du) - I ( w du)

I don't see how this follows from the basic definition of I as supremum
of integrals of simple functions. Can someone please help with a proof?

You got several replies the first time you posted this question.

It's not his fault. Google is acting up, and the search function is
missing lots of posts (can't figure out which ones it is missing). If
he is doing a search/advanced search for his posts to see if anybody
followed up, then they may not be showing up as a result (although
they do show up in the extensive listing by newsgroup) and so he
thinks they disappeared and posts again.

I don't really know what it is that google search is finding or not
finding right now. It's not just the messages made through google,
though.


For example, searching for "Ullrich" on the last 24 hours gives only
three results (none of them related to you: one in Team Endorphin, one
in de.comp.lang.c, one in rec.bycicle.racing


..... Hmmm. It seems it may just be taking a long time before the
search function "sees" them. Your messages of May 2nd seem to be
showing up now (as do mine, although they didn't yesterday). Sigh...



--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu
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Google
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