TITLE: Absolute value of Lebesgue integrable function
QUESTION [5 upvotes]: I want to prove that a measurable function $f$ is Lebesgue integrable iff $|f|$ is.
I've proved the first part  but how can I show if $|f|$ is Lebesgue integrable then $f$ is ?

REPLY [8 votes]: Well the both parts are easy.
If $f$ is Lebesgue integrable than $f^+$ and $f^-$ are Lebesgue integrable, so integral of both $f^+$ and $f^-$ is finite. Thus, their sum is also finite, which means, integral of $|f|$ is finite. If $f$ is measurable than $|f|$ is measurable. If you combine last 2 statements, you get that $|f|$ is Lebesgue integrable.
Same in the other direction, if $|f|$ is Lebesgue integrable, $f^+$ and $f^-$ are also, so integral of $f^+$ and integral of $f^-$ are finite, thus their subtraction is finite, which is integral of $f$.
$f^+ = \max\{0,f\}$
$f^- = \max\{0,-f\}$
$f = f^+ - f^-$
$|f|= f^+ + f^-$