I have three sets of functions in $L_{2w}(-1,1)$ with w=1:
$f_1 = x^2$ for all x
$f_2 = x^2$ if x is irrational and zero otherwise
$f_3 = x^2$ if x is rational and zero otherwise
I want to know if those represent the same vector in L2, so what I need is to see if the set they differ has measure zero. I know that the difference between $f_1$ and $f_2$ is $f_3$ and that the set of rational numbers is countable, so it has measure zero, bot does this apply to $x^2$ too? And I think that $f_1$ and $f_2$ have the same measure, but how can this be true if there are irrational numbers whose square is rational?