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Let $u,u_k \in C^{0}(K)$and $u_0 \in C(\partial K)$ where $K \subset \mathbb{R}^{n}$ is a compact set. Assume that $u_k \rightarrow u$ uniformly and exist $x_0, x_1 \in \partial K$such that $u(x_0) < 0$ and $u(x_1)>0$.

  1. These hypotheses are sufficient to guarantee that \begin{equation} \mbox{med}(\{u_k>0\}) \rightarrow \mbox{med}(\{u>0\}) \end{equation} or \begin{equation} \mbox{med}(\{u_k>0\}) \rightarrow \mbox{med}(\{u>0\})? \end{equation}

  2. Is there some type of converge such $\{u_k>0\} \rightarrow \{u>0\}$ or $\{u_k>0\} \rightarrow \{u>0\}?$

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What is $\mathrm{med}$? – Alexander Shamov Aug 21 '12 at 21:21
I think a slight modification of my counterexample still works here. – Alex Becker Aug 21 '12 at 21:22
med is the lebesgue measure. – user29999 Aug 21 '12 at 21:23

A natural sufficient condition is $\mathrm{med}\{u = 0\} = 0$. It follows from weak convergence of the $u_k$-images of Lebesgue measure.

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