Thread: Norm of indicator function in L^2-space

1. Norm of indicator function in L^2-space

I am reading a book by one John Conway, and at one point he seems to imply that $\displaystyle \|\chi_{\Delta}\|_2 = \left(\int |\chi_{\Delta}|^2 d \mu\right)^{1/2} = (\mu({\Delta}))^{1/2}$ (where $\displaystyle \chi_{\Delta}$ is the indicator function).

I can't quite understand this. I feel it should just be $\displaystyle \mu({\Delta})$...

So, does anyone know why?

2. Because $\displaystyle {\chi_\Delta}^2=\chi_{\Delta\bigcap\Delta}=\chi_\D elta, and \int \chi_\Delta d\mu=\mu(\Delta)$

3. Originally Posted by karkusha
Because $\displaystyle {\chi_\Delta}^2=\chi_{\Delta\bigcap\Delta}=\chi_\D elta$
Yes, of course! Thanks!

4. Originally Posted by karkusha
Because $\displaystyle {\chi_\Delta}^2=\chi_{\Delta\bigcap\Delta}=\chi_\D elta, and \int \chi_\Delta d\mu=\mu(\Delta)$
Wait a tick - the square is outside the absolute value, not inside...

5. $\displaystyle |\chi_\Delta|^2=|{\chi_\Delta}^2|=|\chi_\Delta|=\c hi_\Delta$