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(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

eraldcoil
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    Why is the integral of any function over a set of measure $0$ is equal to $0$? There are still parts which have to be explained. And proving this will require to use simple functions. You can't completely avoid them, as they are part of the definition of the Lebesgue integral. – Mark Dec 17 '23 at 15:52
  • But assuming that the Lebesgue integral over a set of measure 0 is 0. Would the proof be valid? – eraldcoil Dec 17 '23 at 15:53
  • Everything you did works. If you already proved all the properties of the integral that you used here of course. – Mark Dec 17 '23 at 15:57

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