Is this space complete?

Question

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does anybody here have an idea how to tackle this problem?

Answer 1

As pointed out in the comments, $d$ will become a metric if we rather consider equivalence classes of functions.

It has been shown in this website that this metric is equivalent to convergence in measure, i.e. $d(f_n,f)\to 0$ if and only if $f_n\to f$ in measure.

Here we can rediscover the fact that if a sequence converges in measure, a subsequence converges almost everywhere.

Take $(f_n)_n$ a Cauchy sequence for $d$; we can construct inductively a positive integer $n_k$ such that $n_k\gt n_{k-1}$ and $d(f_{n_k},f_{n_{k-1}})\leqslant 2^{-k}$. Using the Borel-Cantelli lemma, we can show that for almost every $x$, $(f_{n_k}(x))_{k\geqslant 1}$ is a Cauchy sequence, which converges to some $f(x)$. By Fatou's lemma, $d(f_{n_k},f)\to 0$ as $k\to\infty$. To conclude, we need to show that the whole sequence converges to $f$: we combine the later fact with the assumption that $(f_n)_n$ is a Cauchy sequence.