Example of Converge in measure, but not converge point-wise a.e.?

Question

Can anyone give an exam of Converge in measure, but not converge point-wise a.e.?

And also for the converse part, professor asks us to prove "pointwise a.e. implies converge in measure", but think about this function:

$$f_n(x)= \chi_{[n,\infty)}$$

It converge to $f(x)=0$ pointwise, but it seems that the difference measure between $f(x)$ and $f_n(x)$ is always infinity.

Answer 1

For the first part, consider the typewriter sequence (Example 4).