Let $\displaystyle E=[0,2]$ and $\displaystyle f_{n}= \chi_{[1/n,2/n]}, \ n=1,2,....$ Show that $\displaystyle f_{n}$ converges almost uniformly on E but not uniformly.
To see convergence is not uniform, observe that for every $\displaystyle 0<\epsilon<1$, no matter how large an n you pick, $\displaystyle |f_n|=1>\epsilon$ on $\displaystyle [1/n,2/n]$.
To see convergence is almost uniform, observe that the set above is the only set on which uniform convergence fails. So let $\displaystyle \epsilon>0$ and chuse n so large that $\displaystyle \mu([1/n,2/n]) =: \mu(B) < \epsilon$. Then $\displaystyle f_n$ restricted to [0,2]\B is identically zero, which clearly converges uniformly to 0.