Uniform Convergence

**Chandru1** · April 7th 2010, 07:59 AM

Let $E=[0,2]$ and $f_{n}= \chi_{[1/n,2/n]}, \ n=1,2,....$ Show that $f_{n}$ converges almost uniformly on E but not uniformly.

**maddas** · April 8th 2010, 02:43 PM

To see convergence is not uniform, observe that for every $0<\epsilon<1$ , no matter how large an n you pick, $|f_n|=1>\epsilon$ on $[1/n,2/n]$ .

To see convergence is almost uniform, observe that the set above is the only set on which uniform convergence fails. So let $\epsilon>0$ and chuse n so large that $\mu([1/n,2/n]) =: \mu(B) < \epsilon$ . Then $f_n$ restricted to [0,2]\B is identically zero, which clearly converges uniformly to 0.

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