The theorem and its proof and proposition 9 that is mentioned in the proof are given below:
My question is:
I do not understand the last part in the last sentence of the proof "by proposition 9, the integral of a nonnegative function over a set of measure zero is zero", I do not understand how proposition 9 leads to this understanding, could anyone explains this for me, please?
If $E$ has measure $0$ the any measurable function is $0$ almost everywhere on $E$. Because the statement $f(x)=0$ if $ x\in E$ and $x \notin E$ is vacuously true.