3
$\begingroup$

Can you interchange limits and supremums of functions?

That is, does $\lim_{n \to \infty} \sup_{x \in X} f_n(x) = \sup_{x \in X} \lim_{n \to \infty} f_n(x) ?$

Thank you!

3 Answers 3

5

No, in general you cannot do that. Imagine if $X$ is $\mathbb{R}$ and $f_n(x)$ is zero except on $[n,n+1]$ where it is $1$. Work out both sides of the equation in this case...

  • 0
    Thanks, awesome counterexample :D2011-11-20
3

No. Consider, for example $f_n(x) = \frac{1}{(x-n)^2+1}$ and $X=\mathbb R$. The supremum of each $f_n$ (and thus the limit of the suprema) is $1$, but the pointwise limit at each $x$ (and thus the supremum of the limits) is $0$.

You'll have better luck if you can assume that the $f_n$s converge uniformly on $X$.

  • 0
    Thanks! Yeah I guess uniform convergence is a lot more $p$owerful.2011-11-20
3

$f_n(x)\leq \sup_{x\in A} f_n(x) $ $\Rightarrow {\liminf}_{n\rightarrow \infty} f_n(x)\leq {\liminf}_{n\rightarrow \infty}\sup_{x\in A}f_n(x) $ $\Rightarrow \sup_{x\in A}{\liminf}_{n\rightarrow \infty} f_n(x)\leq {\liminf}_{n\rightarrow \infty}\sup_{x\in A}f_n(x) $