Show that:
|f - g| is integrable.
If $\displaystyle h_1,h_2$ are integrable then $\displaystyle h_1\pm h_2$ are integrable.
And if $\displaystyle h_3$ is integrable then $\displaystyle |h_3|$ is integrable.
Therefore, if $\displaystyle f,g$ are integrable then $\displaystyle f-g$ is integrable and that implies $\displaystyle |f-g|$ is integrable.