# Estimate on the difference between the measure of the sublevels of two functions in terms of their $L^1$ distance

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 $$\Vert f-g\Vert_{L^1}$$?

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