Sampling rate for a real signal, which is not band-limited?
The power of signals and noise below the Nyquist frequency is properly recovered after sampling. The power of the remainder of the signal (and noise) above the Nyquist frequency is folded into the base band and is usually regarded as interference (more noise): - 



So, after sampling you have a signal to noise ratio and that SNR can be improved by using an anti-alias filter. What remains to be asked is: -


Is the signal below Nyquist sufficiently representative of what you want so that it is usable?
Is the noise sufficiently low for your purposes?