subadditivity of complex numbers on the real line

Question

I am trying to prove that an integral given by

\begin{equation} \int^{\infty}_{-\infty} |f|^2 dx \end{equation} diverges where $f$ is complex.

I can show easily that \begin{equation} |\int^{\infty}_{-\infty} f^2 dx| \end{equation} diverges.

I want to know if \begin{equation} |\int^{\infty}_{-\infty} f^2 dx|\leq \int^{\infty}_{-\infty} |f|^2 dx \end{equation} on the real line.

I looked at the wikipedia page on the absolute value, and it just said that it held for a measurable subset $E$. (https://en.wikipedia.org/wiki/Absolute_value) Unfortunately, I am a big lost regarding what a measurable subset is.