Why are we always interested in mean-squared distortion?
Consider a band-limited function $u(t)$ sampled at an appropriate sampling rate $f_s=1/T$ such that it is perfectly represented by its samples $u(kT)$. If those samples are quantized resulting in values $v(kT)$ then it can be shown that the mean-squared error

$$\epsilon=\sum_k\big|u(kT)-v(kT)\big|^2$$

is equal to the mean-squared error between the waveforms

$$\epsilon=\frac{1}{T}\int_{-\infty}^{\infty}\big|u(t)-v(t)\big|^2dt$$

where $v(t)$ is the waveform reconstructed from the quantized samples $v(kT)$. So quantizing in such a way that the mean-squared error between the original and the quantized samples is minimized, guarantees that also the mean-squared error between the corresponding continuous waveforms is minimized. This is not the case with other error measures.