$f_n\to f$ in a metric VS $f_n\to f$ in measure.

Question

I have a function $\rho(f,g)$ to be the metric function for any two measurable functions $f,g$.

What does it mean by $f_n\to f$ in a metric and $f_n\to f$ in measure, where $(f_n)$ is a sequence of functions.

Answer 1

$f_n \to f$ in the metric $\rho$ means that $\lim_{n \to \infty} \rho(f_n, f)=0$.

$f_n \to f$ in the measure $\mu$ means that for all $\varepsilon >0$ you have $\lim_{n \to \infty} \mu ( \{ x : |f_n(x) - f(x)| > \varepsilon\} ) = 0$