Fix $R\gg 1$. How can I estimate the difference between the Lebesgue measures $$\mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): f(x)>0\} - \mathscr{L}^N(\{x \in \mathbb{R}^N \cap B_R(0): g(x)>0\})$$ in terms of the $L^1$ difference between the functions $f$ and $g$, that is $\Vert f-g\Vert_{L^1}$?

  • 2
    You can't: just take $f>0$, $g<0$ with tiny $L^1$ norms. – Christian Remling Nov 24 at 17:44

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