# Local summability - Lebesgue Integration

### Local summability - Lebesgue Integration

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 = $$\frac{a}{|x|^b}$$. 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 ?

Peter
Guest

### Re: Local summability - Lebesgue Integration

Edit : the function is not $$\frac{a}{|x|^b }$$ but simply $$\frac{1}{|x|^b }$$
Guest

### Re: Local summability - Lebesgue Integration

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

### Re: Local summability - Lebesgue Integration

$$\frac{1}{|x|}$$ 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