Problem 1347 (difficulty: 5/10)

Let \(\displaystyle f\) be bounded and non-negative on the measurable set \(\displaystyle A\). Prove that \(\displaystyle \int _A f=0\) implies that \(\displaystyle k(\{ x\in A: f(x)\ge a\})=0\) for all \(\displaystyle a>0\). Is the converse true?

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