Suppose $f_n(x),f(x)$ are both continuous functions with domain $M\subset R^2$ and codomain $R$, and define $L=\{x:f(x)\geq c\}$ and $L_n=\{x:f_n(x)\geq c\}$. Suppose $\underset{x\in M}{sup}|f_n(x)-f(x)|\rightarrow 0$. Do we have convergence of $L_{n}$ to $L$ in the following sense:
$\mu(L\Delta L_{n})\rightarrow 0$, where $\mu(\cdot)$ denotes the Lebesgue measure and $L\Delta L_{n}$ denotes the symmetric set difference: $L\Delta L_{n}=(L\cap L_{n}^c)\cup( L_{n}\cap L^c)$ (in plain words, the Lebesgue measure of the difference of these two sets vanishes)?
Let $f(x)=0$ and $f_n(x)=-1/n$ for all $x\in M$ and all $n$. Let $c=0$. Then $L=M$ but $L_n=\phi$, and $\mu(L\Delta L_n)$ does not converge to $0$. (Unless $\mu(M)=0$.)