Factor $|a|^{-1/2}$ in definition of mother wavelets
My answer is for real scale $a$ and the fact that wavelet transform is usually defined in $L_2$ with norm

$$||\Psi(\tau)|| = \int_\mathbb{R} \Psi(\tau)\Psi^*(\tau)\mathrm{d}\tau $$

So

$$||\Psi_{a,t}(\tau)|| = \int_\mathbb{R} \frac{1}{|a|}\Psi(\frac{\tau-t}{a})\Psi^*(\frac{\tau-t}{a})\mathrm{d}\tau$$

Set $\tau' = \frac{\tau-t}{a} \implies d\tau' = d\tau / a \implies d\tau = a \times d\tau'$

$$||\Psi_{a,t}(\tau)|| = \int_\mathbb{R} \frac{1}{|a|}\Psi(\tau')\Psi^*(\tau') \times a \times \mathrm{d}\tau' = \int_\mathbb{R} \Psi(\tau')\Psi^*(\tau') \mathrm{d}\tau' = ||\Psi(\tau)||$$

Update for the question "How does it not change the norm considering that it could introduce a negative sign with the norm being sign-less?"

We develop stuff for the case that scaling factor $a &lt; 0$.

with $A &gt; 0$
$$||\Psi_{a,t}(\tau)|| = \lim_{A \to \infty}\int_{(-A-t)/a}^{(A-t)/a} \frac{1}{|a|}\Psi(\tau')\Psi^*(\tau') \times (-|a|) \times \mathrm{d}\tau' \\
= \lim_{A \to \infty}\int_{(A-t)/a}^{(-A-t)/a} \Psi(\tau')\Psi^*(\tau') \times \mathrm{d}\tau' \\
= \lim_{A \to \infty}\int_{(-A+t)/|a|}^{(A+t)/|a|} \Psi(\tau')\Psi^*(\tau') \times \mathrm{d}\tau' = \int_{-\infty}^{+\infty} \Psi(\tau')\Psi^*(\tau') \times \mathrm{d}\tau'$$