The mathematical definition of the spectrum of a stationary process $\mathbf{x}(t)$is to take the Fourier transform of a finite segment
$$\mathbf{X}_T (u) = \frac{1}{T} \int_{-T/2}^{T/2} \mathbf{x} e^{-2\pi i u t}dt $$,
then after ensemble averaging form the following limit
$S(u) = \lim_{T \to \infty} {\frac{\textbf{Av} \{|\mathbf{X}_T (u)|^2\}}{T}}$
In all practical spectrum analyzers though, equipment such as by Hewlett-Packard (Agilent, Keysight) makes, the following is really done. The output of a bandpass filter whose impulse response is represented by $h(t,u,\delta)=\frac{1}{\delta}\int_{u-\delta/2}^{u+\delta/2}e^{i2\pi vt}dv=e^{i2\pi ut} \textrm{sinc}(\delta t)$ is passed through a square-law detector and then averaged by a so-called video filter that is essentially an integrator:
$$h(t,u,\delta)=\frac{1}{\delta}\int_{u+\delta/2}^{u+\delta/2}\\ \mathbf{y}(t,u,\delta)=\mathbf{x(t)}\otimes h(t,u,\delta)\\ S'(u,A,\delta)=\frac{1}{A}\int_{A/2}^{A/2} dt|\mathbf{y}(t,u,\delta)|^2 $$
My question is in what sense $S'(u,A,\delta)$ is an approximation to $S(u)$ and what is the error? Thank you.