Exemples for strict inequalities in measure theory.

Question

Let $(E_{n})$ a sequence of measurable sets. Then I proved that

(a) $m(\liminf E_{n}) \leq \liminf m(E_{n})$

and

(b) $m(\limsup E_{n}) \geq \limsup m(E_{n})$ when $m(\cup E_{n}) < \infty$.

Now, I'm trying to get examples where the inequalities are strict, but I didn't succeed. Could anyone give me some examples? Also, I would like to know what conditions guarantee equality, is necessary the convergence?

Answer 1

Let $E_n=(0,1)$ for $n$ even and $E_n=(1,2)$ for $n$ odd. Then $\lim \inf E_n$ is empty so strict inequality holds in a). Can you do a similar thing for 2) ? Hint: see if the same example works!