$|f| $ is Lebesgue integrable , does it implies $f$ is also? If $ f $ is Lebesgue integrable then $|f|$ is Lebesgue integrable but does the converse of the result is also true?

Think of |f| as a division of f into two functions: $f_+$ and $f_-$.
$f_+$ we define as equal to f on the domain {x: f(x) is non-negative}, and 0 on all other x.
$f_-$ we define as equal to -f on the domain {x: f(x) is negative} and 0 elsewhere. $|f|=f_+ + f_-$.
If |f| is finite, then necessarily both $f_+$ and $f_-$ are finite.
