Limits in measure on a non-$\sigma$-finite measure space $(X, \mathcal{A} , \mu)$ need not be unique.
May someone explain me why, better with an example?
Is it possible that, in the case of non $\sigma$-finite measure space, the convergence almost everywhere doesn't imply the convergence in probability?
Here is an example of a non $\sigma$-finite measure space, the convergence almost everywhere doesn't imply the convergence in probability. Consider $(\Bbb R, \mathscr P(\Bbb N),\alpha)$, where $\alpha$ is the counting measure. Since $\Bbb R$ is uncountable, $\alpha$ is not $\sigma$-finite. Let $$f_n(x)=\begin{cases} 0 &\text{if } x\le n,\\ 1 &\text{if } x >n. \end{cases}$$ Then $\{f_n\}_n$ converges to $0$ everywhere. However, $\{f_n\}_n$ does not converge to $0$ in probability.
In fact, the convergence almost everywhere doesn't necessarily imply the convergence in probability for a $\sigma$-finite measure. For example, $\{f_n\}_n$ does not converge to $0$ in probability with respect to the Lebesgue measure on $\Bbb R$, either.