Convergence in measure: two definitions

Question

We say that $f_n$ converges to $f$ in measure if for all $\epsilon > 0$ there exists $N$ such that if $n > N$ then $\mu \{x: \mid f(x) - f_n \mid > \epsilon\} < \epsilon$.

Is it equivalent to requiring that for each $\epsilon >0$ we have $\lim \mu \{x: \mid f(x) - f_n \mid > \epsilon\} = 0$?

Thank you

Answer 1

Note that for $\varepsilon'>\varepsilon>0$ we have $$\{x: |f(x)-f_n(x)|>\varepsilon'\} \subseteq \{x: |f(x)-f_n(x)|>\varepsilon\}$$ so, $$\mu(\{x: |f(x)-f_n(x)|>\varepsilon'\}) \leq \mu(\{x: |f(x)-f_n(x)|>\varepsilon\})<\varepsilon$$

Then, the first statement implies the second. I left the details to you.