4
$\begingroup$

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 ?

  • 1
    It cannot be. If $A$ is a non measurable set in $[0,1]$, then define $f = -1 +2(1_A)$. Then $|f|=1$ and isintegrable, but $f$ is not.2012-10-08
  • 2
    Maybe you are looking to add the hypothesis that $f$ is measurable function into the (possibly extended) reals?2012-10-08
  • 4
    Look at $f^+$ and $f^-$2012-10-08
  • 0
    @leo's comment hold the key...2012-10-08

2 Answers 2