Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such that $N_0(f) < + \infty.$
We know that $L^0$ is a vector space and $N_0$ is a pseudo-norm on $L^0$.
My question is: how we can prove that $L^0$ is a complete metric space?