Proof of a CTFT property
If $f(t)$ is a real non-negative function with Fourier transform

$$F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\tag{1}$$

the following must hold:

$$|F(\omega)|=\left|\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\right|\le\int_{-\infty}^{\infty}|f(t)|dt=\int_{-\infty}^{\infty}f(t)dt=F(0)$$

since $f(t)=|f(t)|$.