Let Ω⊂Rn a non empty open set and f:Ω→R a nonnegative measurable function with ∫Ωf=0. Then f=0 in Ω almost everywhere.
I have no idea of how to start this problem, someone could help me ?
Thanks in advance!
My try (I am not sure):
Let En:={x∈Ω;f(x)>1/n},n∈N and define E:={x∈Ω;f(x)>0}=∪n≥1En.
Note that
0=∫Ωf≥∫Ef≥∫Enf≥|En|n≥0.
Then |En|=0 for all n, which implies |E|=0. Then f=0 in Ω a.e
I am not sure because it seems that we can replace the set Ω by a measurable set with zero measure and if we consider a set like this the affirmation is not true.
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