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 ?
Absolute value of Lebesgue integrable function
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real-analysis
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1It 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
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2Maybe you are looking to add the hypothesis that $f$ is measurable function into the (possibly extended) reals? – 2012-10-08
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4Look at $f^+$ and $f^-$ – 2012-10-08
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0@leo's comment hold the key... – 2012-10-08