A signal $g(t)$ is passed through a squaring device and then through an LPF such that bandwidth of the LPF tends to 0
I don't think that using a Dirac impulse really helps to see what's going on in this problem. Using the notation of the given solution we have

$$\mathcal{F}\left\{g^2(t)\right\}=A(\omega)\;\Longrightarrow\;A(0)=E_g$$

and

$$Y(\omega)=A(\omega)H(\omega)$$

where $H(\omega)$ is the frequency response of the ideal low pass filter.

The output signal is given by

$$\begin{align}y(t)&amp;=\frac{1}{2\pi}\int_{-\infty}^{\infty}Y(\omega)e^{j\omega t}d\omega\\&amp;=\frac{1}{2\pi}\int_{-\infty}^{\infty}A(\omega)H(\omega)e^{j\omega t}d\omega\\&amp;=\frac{1}{2\pi}\int_{-2\pi\Delta f}^{2\pi\Delta f}A(\omega)e^{j\omega t}d\omega\\&amp;\approx\frac{1}{2\pi}A(0)\cdot 4\pi\Delta f\\&amp;=2E_g\Delta f\end{align}$$

with the approximation being justified for $\Delta f\to 0$.