# A L^2(0,1) space function

1. Feb 14, 2017

### LagrangeEuler

If some function is element of space $L^2(0,1)$ then
$$\int^1_0|f(x)|^2dx< \infty$$. What in the case when it is not so simple to calculate this integral. For example $f(x)=x^{-1}(C_1+C_2 \ln x)$. How to find is it this function in $L^2(0,1)$ for some $C_1,C_2$?

2. Feb 14, 2017

### mathman

It is never (other than both constants = 0). Integral diverges at x=0.

3. Feb 26, 2017

### Amal Chacko

4. Feb 26, 2017

### mathman

I have never seen the term Lebesgue differentiation. Lebesgue integration is a theory of integration based on measure theory.

5. Feb 26, 2017

### WWGD

This is not quite the same as Lebesgue differentiation, bit it is kind of close to it:

https://en.wikipedia.org/wiki/Lebesgue's_density_theorem

6. Mar 8, 2017

7. Mar 9, 2017

### WWGD

Ah, yes, Hawkeye's link is clearly better, more direct than mine.