Saturday, 20 May 2017

measure theory exercise: null integral implies null function

Let ΩRn a non empty open set and f:ΩR 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:={xΩ;f(x)>1/n},nN and define E:={xΩ;f(x)>0}=n1En.



Note that



0=ΩfEfEnf|En|n0.



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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