B. P. Lathi in his book "Principles of Linear Systems and Signals" mentions in the Fourier Transform:
When $x(t)$ is periodic, the spectrum is discrete, and $x(t)$ can be expressed as a sum of discrete exponentials with finite amplitudes. However for an aperiodic signal, the spectrum becomes continuous; that is, the spectrum exists for every value of $\omega$, but the amplitude of each component in the spectrum is zero. The meaningful measure here is not the amplitude of a component of some frequency but the spectral density per unit bandwidth.
I get that when the period of the periodic signal tends to infinity, the Fourier coefficient of the Fourier series tends to zero, at least mathematically makes sense. But, physically, seeing all the continuous spectra of a Fourier transform, this amplitude of each component in the spectrum is zero makes no sense. I can literally see an amplitude. At a particular frequency. How is it zero????? What am I missing?