I have certain doubts about convergence in measure.
We know that, for example, $f_n = \Bbb{1}_{(0,n)}\frac{1}{n} $ converges in Lebesgue measure to zero. That is : $ \lim_{n \to \infty} \lambda( x \in \Bbb{R}: \vert f_n(x) - f(x) \vert \gt \alpha ) = 0 $ for every $\alpha \geq 0 $
Now, my doubts begin. Take $\alpha = \frac{1}{9}$
A set that satifies $\vert \Bbb{1}_{(0,n)} \frac{1}{n}\vert \gt \frac{1}{9} $ is the set $(0,8)$ with Lebesgue measure 8. If we take the limit its not zero.
Now, of course, I am confusing something. I would like to know what is my mistake. Thanks.
I think you are mixing the variables $n$ and $x$.
If $n \geq 9$ then $|1_{(0,n)} (x)\frac 1n|\leq |\frac 1n| \leq \frac 1 9$ for any $x$. So, $|1_{(0,n)} (x)\frac 1n|>\frac 19$ cannot hold for any $x$ if $n \geq 9$.