Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it true that: \begin{equation} argmin_{x \in X} (f+g) = \lim_{n\mapsto \infty} argmin_{x \in X} (f_n+g_n)\,? \end{equation}
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1$\begingroup$ You have to give more details on the convergence $f_n\to f$. Also what do you mean by $\DeclareMathOperator{\amin}{argmin}$ $\amin_{x\in\mathbb{R}}\sin x$? $\endgroup$– Liviu NicolaescuCommented May 16, 2016 at 17:25
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$\begingroup$ $argmin_{x\in \mathbb{R}}sin(x)=\{\frac{\pi}{2}+k\pi:k \in \mathbb{N}\}$... I never assumed the argmin function took unique (or non-empty) values.... $\endgroup$– ABIMCommented May 17, 2016 at 12:42
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$\begingroup$ With only pointwise convergence I doubt you can say something meaningful. Also you need to define the motion of convergence of sets that you have in mind since, according your comment argmin is a st. $\endgroup$– Liviu NicolaescuCommented May 17, 2016 at 18:31
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1$\begingroup$ You should look up the notion of Gamma convergence. $\endgroup$– DirkCommented Jul 13, 2016 at 14:06
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$\begingroup$ I did a few days after posting this; that's exactly what I used since my functions were l.s.c. $\endgroup$– ABIMCommented Jul 13, 2016 at 15:31
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