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Let $\{u_{j}\}_{j\in\mathbb{N}}$ be a bounded sequence in $L^{\infty}(\Omega)$ for a given smooth bounded domain $\Omega \subset \mathbb{R}^{n}$. Assume $u_{j} \to u\in L^{\infty}(\Omega)$ a.e. in $\Omega$. So, my question is, is it true in general that $\lim\limits_{j\to\infty}\min\limits_{x\in B(x,r)}u_{j}(x)=\min\limits_{x\in B(x,r)}u(x)$ and $\lim\limits_{j\to\infty}\max\limits_{x\in B(x,r)}u_{j}(x)=\max\limits_{x\in B(x,r)}u(x)$ in general?

if $u_{j}\in C(\Omega)$, then I believe it is quite obvious but $L^{\infty}(\Omega)$ I am not sure.

Any help will be appreciated! Thank you very much!

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No. Let $\Omega = (0,1) \subseteq \mathbb{R}$. Let $u \equiv 1$. Let $u_n = 1$ on $(\frac{1}{n},1)$ and $u_n = 0$ on $(0,\frac{1}{n})$. Then $u_n \to u$ pointwise everywhere, $\min_{x \in \Omega} u_n(x) = 0$ for each $n$, and $\min_{x \in \Omega} u(x) = 1$.

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  • $\begingroup$ Thank you very much for your answer! It helps me a lot! $\endgroup$ – Evan William Chandra Sep 20 '18 at 7:33

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