(From Bartle)
Query, I have seen solutions to this problem using simple functions, etc. but is it possible to solve it like this? I put my solution
Solution. $f(x)=0$ almost x then the set $E=\left\{x\in X:f(x)\neq 0\right\}$ has a measure $0$. Then, $\int_X f=\int_{E}f+\int_{E^c}f=0+0=0$.
Does this argument work? or is it insufficient? thank you