One way to think about this is in the frequency domain in terms of filtering.
The measuring apparatus has in effect a low-pass filter at the input with a corner frequency far below $\omega_0$. No physical measuring apparatus has infinite bandwidth and thus, there is a high-frequency "roll-off" in the frequency response of the instrument..
The square of a sinusoid contains a DC component and a component at twice $\omega_0$:
$\cos^2(\omega_0 t) = \frac{1}{2}[1 + \cos^2 (2\omega_0 t)]$$\cos^2(\omega_0 t) = \frac{1}{2}[1 + \cos (2\omega_0 t)]$
but the only component that is measured is the DC component since the high-frequency component is "filtered" out.