Pointwise convergence vs convergence in measure

Question

I would be very grateful if I could be given an example of a sequence convergent pointwise but it is not convergent in measure and an example of a sequence convergent in measure but it is not convergent pointwise.

Answer 1

$I_{(n,\infty)}$ converges at every point to $0$ in the real line but the Lebesgue measure of the set $\{x:I_{(n,\infty)}(x)>1\}$ is $\infty$ for every $n$ so the sequence does not converge in measure.

On $(0,1)$ with Lebesgue measure consider the intervals $[\frac {i-1} {2^{n}},\frac i {2^{n}}), 1\leq i\leq 2^{n}, n\geq 1$. Arrange this in a sequence $(E_n)$, say with increasing values of $n$. Then $(I_{E_n})$ converges in measure but does not converge at any point. (At every point the sequence contains infinitely many $0$'s and infinitely many $1$'s)