Let $(X,S,\mu)$ be a measure space with $\mu(X)<\infty$ and define for each $f\in\mathbb{M}(X,S)$ ($f$ measurable):
$$r(f):=\int\frac{|f|}{1+|f|}d\mu$$
I showed that:
$r(f)<\infty$
$d(f,g):=r(f-g)$ is a metric
$d(f_n,f)\xrightarrow[]{n\to\infty}0\Leftrightarrow f_n\xrightarrow[\mu]{n\to\infty}f$ (Pointwise convergence iff convergence in measure)
Now I would like to demonstrate completeness. I want to show that if $(f_n)$ is a Cauchy sequence with respect to $d$ in $\mathbb{M}(X, S)$, then there exists $f\in\mathbb{M}(X,S)$ such that $d(f_n,f)\xrightarrow[]{n\to\infty}0$.
I think the result is a simple consequence of the F. Riesz - H. Weyl theorem but I'm not entirely sure.