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 $f^-$.