Prove that $f$ is bounded in probability iff $f$ is finite a.e.
We say that $f$ bounded in probability if given $\epsilon > 0$, there exists a finite real number $M_{\epsilon}$ such that $m(\{|f| \leq M_{\epsilon}\}) \geq 1 - \epsilon$.
- $m$ is the Lebesgue measure.
I'm trying to prove the first implication. My approach is take $\{|f| = \infty\}$ and show that $m(\{|f| = \infty\}) = 0$, but, I don't know how to use the hypothesis with this ideia. Maybe, there is a some simpler way. Can someone help me?
Hint For each $M>0$ we have $$m(\{ |f| =\infty\}) \leq m(\{ |f| >M \})$$