Local summability - Lebesgue Integration

Local summability - Lebesgue Integration

Postby Guest » Wed May 26, 2021 1:06 pm

Hi everyone,

I have been studying Lebesgue integration for few weeks now and I cannot understand its philosophy ...

I started to look at simple exercises to begin with, such as the following one :
Let f = [tex]\frac{a}{|x|^b}[/tex]. The question is "for what values of b is f locally summable (or integrable) ?"

I have plotted lots of different possibilities and I really cannot see why it is not integrable over the whole set of real numbers.

What am I missing ?

Thanks in advance for any help you can provide.
Peter
Guest
 

Re: Local summability - Lebesgue Integration

Postby Guest » Wed May 26, 2021 1:08 pm

Edit : the function is not [tex]\frac{a}{|x|^b }[/tex] but simply [tex]\frac{1}{|x|^b }[/tex]
Guest
 

Re: Local summability - Lebesgue Integration

Postby Guest » Mon Jun 07, 2021 8:05 am

What definition of "locally summable" are you using?
Guest
 

Re: Local summability - Lebesgue Integration

Postby Guest » Fri Jun 18, 2021 12:54 pm

[tex]\frac{1}{|x|}[/tex] is "locally summable" for all x except x= 0.

Do you see why, for some values of b, there would be a problem at x= 0?
Guest
 


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