My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
December 2nd, 2012, 04:12 PM   #1
Newbie
 
Joined: Dec 2012

Posts: 9
Thanks: 0

LEBESGUE INTEGRATION

Does anyone know a function belonging to L^p, but not to L^q, for all q<p? And a function belonging to L^p but not to L^q, for all q>p?. "p" is a real number which is arbitrary but fixed. I think these functions exist, but I am not able to find one of them. Thanks.
thamy_271091 is offline  
 
December 3rd, 2012, 01:27 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,628
Thanks: 622

Re: LEBESGUE INTEGRATION

You need to define the domain of integration.
mathman is offline  
December 3rd, 2012, 03:04 PM   #3
Newbie
 
Joined: Dec 2012

Posts: 9
Thanks: 0

Re: LEBESGUE INTEGRATION

Well, you are right, but I did not define it because you can choose the domain of integration, as long as it works.
thamy_271091 is offline  
December 4th, 2012, 01:13 PM   #4
Global Moderator
 
Joined: May 2007

Posts: 6,628
Thanks: 622

Re: LEBESGUE INTEGRATION

Example 1/?x
(interval [0,1]). Belongs to Lp for p < 2, not Lq for q ? 2.
(interval [1,?). Belongs to Lp for p >2, not Lq for q ? 2.
mathman is offline  
December 5th, 2012, 06:06 AM   #5
Newbie
 
Joined: Dec 2012

Posts: 9
Thanks: 0

Re: LEBESGUE INTEGRATION

That' s not exactly what I asked. I could explain it better, but I am not sure how to use mathematical symbols here. I know how to use LaTeX, however, I don't know how to use it here.
thamy_271091 is offline  
December 5th, 2012, 10:49 AM   #6
Math Team
 
Joined: Sep 2007

Posts: 2,409
Thanks: 6

Re: LEBESGUE INTEGRATION

Begin with the HTML tag [ t e x ] and end with [ / t e x ]- without the spaces, of course.
HallsofIvy is offline  
December 6th, 2012, 05:22 AM   #7
Newbie
 
Joined: Dec 2012

Posts: 9
Thanks: 0

Re: LEBESGUE INTEGRATION

Thanks for your help. I explain it better,
Let be Lebesgue measure on . Let . Find a function in but that fails to be in for . Find another function in that fails to be in for
thamy_271091 is offline  
December 6th, 2012, 02:05 PM   #8
Global Moderator
 
Joined: May 2007

Posts: 6,628
Thanks: 622

Re: LEBESGUE INTEGRATION

Just to be clear, the domain of integration is the whole real line?

If so, just use the examples I gave where the functions = 0 outside the original intervals. The only difficulty is that Lp would be on the divergent side of the interval, i.e. divergent for q ? p or p ? q for these examples.
mathman is offline  
December 7th, 2012, 07:13 AM   #9
Newbie
 
Joined: Dec 2012

Posts: 9
Thanks: 0

Re: LEBESGUE INTEGRATION

That's precisely the difficulty. For instance, if I choose the function defined on (0,1), that function does not belong to but it belongs to , for all q>p. I was looking for the opposite property. I have finally found an example.

Let be a sequence converging to p. For every we can find a function such that,

(a) and , for any q (For example )

(b) (just multiply by a constant).

We define,

thamy_271091 is offline  
December 7th, 2012, 07:34 AM   #10
Newbie
 
Joined: Dec 2012

Posts: 9
Thanks: 0

Re: LEBESGUE INTEGRATION

That function is an example of a function belonging to but not to , for any q<p. Based on a similar way of reasoning, you can construct a function g belonging to but not to for any q>p. Extending these 2 function to and considering their sum, we can even find a function belonging to but not to , for any q different to p.

PS: I wanted to edit my last post because I did not press submit intentionally but I don't know how.
thamy_271091 is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
integration, lebesgue



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Lebesgue integration question Fabion Real Analysis 0 February 24th, 2014 04:39 AM
is it a lebesgue zero set in R^2? wuspengret Calculus 3 January 15th, 2013 12:54 AM
Lebesgue Integral problem Real Analysis 2 December 2nd, 2009 08:36 PM
Lebesgue Integral problem Real Analysis 1 August 21st, 2009 02:48 PM
lebesgue measure danitg Real Analysis 0 May 10th, 2009 09:01 AM





Copyright © 2018 My Math Forum. All rights reserved.