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Let $X = [0,1]$. Define $f:X\to\mathbb{R}_{\geq 0}$ to be Lipschitz continuous on $X$. Put $Y\subset X:\int\limits_Y f(x)\,dx = 0$ What can we say then about $A = X\setminus Y$? It is not defined uniquely, but I am interested in some common properties of such sets. Namely, can they be nowhere dense, or with boundary of positive measure?

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    If $f$ is an arbitrary nonnegative Lipschitz function, then $Y$ is an arbitrary union of a closed set and a set of measure zero (in the way indicated in Alexander Thumm's comment).2011-06-12

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