There are explicit examples of this. We can have ever thinner disturbances to convergence that keep getting less and less area, but never cease to move around.

If it does converge pointwise to something, then it must match of course.

- Fourier basis is complete for $L_{2}$ | 62, 246, 2
- Riesz-Fischer theorem | 59, 306, 4
- Lebesgue integral of $L_{p}$ is complete but Riemann isn't | 91, 805, 11
- Lebesgue integral | 31, 1k, 15
- Calculus | 17, 7k, 159
- Mathematics | 17, 13k, 329
- Ciro Santilli's Homepage | 262, 181k, 3k

- Fourier basis is complete for $L_{2}$ | 62, 246, 2