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Let $\{f_n\}$ be a sequence of continuous functions defined on $\mathbb{R}$. Give an example of such a sequence that:

  1. converges in measure but nowhere pointwise;
  2. converges almost uniformly but not uniformly.

As a measure, the Lebesgue measure is used.

Although I'm familiar with the above convergence concepts and have used them in the past, I'm having difficulties constructing such examples myself. Any ideas will be greatly appreciated.

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  • $\begingroup$ Sliding bump functions. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Sep 28 '18 at 15:41
  • $\begingroup$ @GNUSupporter8964民主女神地下教會 could you elaborate a bit on your example, or perhaps recommend me some reference material? $\endgroup$ – Chris Sep 28 '18 at 15:45
  • $\begingroup$ The functions used in math.stackexchange.com/q/2611505/290189 $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Sep 28 '18 at 15:48
  • $\begingroup$ Unfortunately, these functions are not continuous $\endgroup$ – Chris Sep 29 '18 at 22:24
  • $\begingroup$ You may read zhw's answer whose second part originates from Fourier series, whose graphs are much more elegant than mine. They give a really nice pictures and a concise proof. It's much better than my answer. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Sep 29 '18 at 22:29

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