Let and Show that converges almost uniformly on E but not uniformly.
To see convergence is not uniform, observe that for every , no matter how large an n you pick, on .
To see convergence is almost uniform, observe that the set above is the only set on which uniform convergence fails. So let and chuse n so large that . Then restricted to [0,2]\B is identically zero, which clearly converges uniformly to 0.