Is an in measure limit of a uniformly bounded sequence bounded?

If $\{f_n\}$ a sequence of measurable functions that are uniformly bounded and if $f_n\rightarrow f$ in measure (in probability), then is $f$ bounded?

No. Take $f_n=0$ and $f(x)=0$ when $x>0$ is irrational, $f(x)=1/x$ when $x>0$ is rational, $f(0)=0$. Then $P(|f_n-f|>\epsilon)=0$ but $f$ is clearly unbounded.