The Wiki article on Wavelets contains this:
$\int_{-\infty}^{\infty} \psi (t)\, dt = 0$ is the condition for zero mean, and
$ \int_{-\infty}^{\infty} |\psi (t)|^2\, dt = 1$ is the condition for square norm one.
Why put an absolute value in the second equation, given that squaring makes the sign positive anyway?
EDIT: $\psi$ is in $L_1(\mathbb{R})\cap L_2(\mathbb{R})$