= Solution
Consider a narrow bundle of rays between two surface elements $dA_1,dA_2$ separated by $s$ in a transparent, stationary Euclidean vacuum. Let the ray make angles $\theta_1,\theta_2$ to the respective surface normals. The receiving element subtends <solid angle> $d\Omega_1=\cos\theta_2dA_2/s^2$ at the emitter. By the definition of <specific intensity>, the beam power in frequency interval $d\nu$ is
$$
d\mathcal P_\nu=I_{\nu,1}\cos\theta_1dA_1\,d\Omega_1\,d\nu
=I_{\nu,1}\frac{\cos\theta_1\cos\theta_2dA_1dA_2}{s^2}\,d\nu.
$$
At the receiving end the source subtends $d\Omega_2=\cos\theta_1dA_1/s^2$, so the identical power is
$$
d\mathcal P_\nu=I_{\nu,2}\frac{\cos\theta_1\cos\theta_2dA_1dA_2}{s^2}\,d\nu.
$$
In the absence of absorption, emission or frequency shifts, <conservation of energy> equates these expressions:
$$
\boxed{I_{\nu,2}=I_{\nu,1}.}
$$
This <vacuum conservation of specific intensity> also follows directly from the <radiative transfer equation> $dI_\nu/ds=0$. The apparent <solid angle> of an unresolved source shrinks as $s^{-2}$, so its integrated observed <radiative flux> still follows the inverse-square law. Specific intensity and unresolved flux are different quantities.
The vacuum and frequency assumptions are essential: absorption and emission change <specific intensity>, and gravitational or cosmological redshift conserves $I_\nu/\nu^3$ along a ray instead of $I_\nu$ itself. A fixed telescope aperture observing an unresolved object also cannot infer constant total received power from intensity conservation.
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