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How do I use this the following result

if $f$ is a non-negative measurable function on $X$, then $\int_X f~d\mu =0$ if and only if $f=0$ a.e. on $X.$

to prove that

if $f$ be an integrable function over $X$, then $\int_E f~d\mu =0$ for every measurable subset $E$ of $X$ if and only $f=0$ a.e. on $X$.

In general how does one approach these types of proof where one proves the result for $f\ge 0$ and apply the result to $f^+$ and $f^-$.

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    You need to notice first that you are decomposing your given function in two others wich are both measurable, because you are taking inverse images of intervals of the form $[0,\infty)$ and $(-\infty,0]$. I'm not sure if I get wat you are asking, so I would like to ask you if you can formulate your question a bit more precisely.2012-05-01
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    @matgaio: Done.2012-05-01

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