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I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of simplicity.

The problem I'm having is that almost everywhere means that the set of values where $f_n$ does not converge to $0$ has measure $0$. But this just seems like a special case of almost uniform convergence which means that we can remove a set of arbitrary small measure from the domain and we will have uniform convergence on the remainder.

So is there some simple example that illustrates the difference between these two modes of convergence?

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  • $\begingroup$ Isn't it the same difference that there is between convergence and uniform convergence ? $\endgroup$ – Ant Aug 11 '14 at 18:16
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Take $f=1$ and $f_n=\chi_{[-n,n]}$

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  • $\begingroup$ Ok, I can see that the set of values where $f_n$ does not converge to $f$ has value $0$ because it actually does converge to $f$ as $n \to \infty$. But why does it not have almost uniform convergence? $\endgroup$ – sonicboom Aug 11 '14 at 18:21
  • $\begingroup$ Measure of the sets where $|f-f_n| \geq1/2$ say is infinite for all $n$ . Hence you do not have a.u. convergence. $\endgroup$ – voldemort Aug 11 '14 at 18:26

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