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.
for the first part :take $f_n$ on $[0,1]$ as $1_{E_{n}}$ and $E_{n}$are like this sequance
$E_{1}= [0,\frac{1}{2}) E_{2}= [\frac{1}{2},1]$
$E_{3}= [0,\frac{1}{3}) E_{4}= [\frac{1}{3},\frac{2}{3}] E_{5}= [\frac{2}{3},1]$
...
and for your other qustion the theorem is true whene $f_{n}$ has finite domain.
For the first part, consider the typewriter sequence (Example 4).
