Let $D$ be a bounded domain in $\mathbb{R}^{N}$ ($N\geq2$) and $E$ a closed subset of $D$ with empty interior. Suppose $f$ is a measurable function defined on $D$ and integrable on $D\setminus E$, i.e., $$\int_{D\setminus E}|f(x)|dx<\infty.$$ Can we say that at least either its positive part $f^{+}$ or its negative part $f^{-}$ are integrable on $D$?
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2$\begingroup$ Only if $E$ had measure zero, and that certainly need not be the case. Just like the "fat Cantor set" construction, you can certainly find closed sets in $\mathbb{R}^N$ with empty interior and positive measure. $\endgroup$– Nate EldredgeCommented Oct 4, 2017 at 2:25
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