December 2nd, 2012, 03:12 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.

December 3rd, 2012, 12:27 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,707 Thanks: 674  Re: LEBESGUE INTEGRATION
You need to define the domain of integration.

December 3rd, 2012, 02:04 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.

December 4th, 2012, 12:13 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,707 Thanks: 674  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. 
December 5th, 2012, 05:06 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.

December 5th, 2012, 09:49 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.

December 6th, 2012, 04:22 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 
December 6th, 2012, 01:05 PM  #8 
Global Moderator Joined: May 2007 Posts: 6,707 Thanks: 674  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. 
December 7th, 2012, 06:13 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, 
December 7th, 2012, 06:34 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. 

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