TITLE: What is a practical use for this metric?
QUESTION [18 upvotes]: Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric
That post has a more generalized form of a metric I occasionally see $d(x,y) = \frac{|x-y|}{1+|x-y|}$. When would using this metric be useful exactly? It occasionally comes up when I study analysis, but I don't know why, I don't know what people use it for or what benefit it could ever bring over the standard metric. I've merely only seen it as an example of a metric in books or sites, but if so many sources mention it, then it's very unlikely that it's useless.

REPLY [2 votes]: I know this thing: if $(X, d)$ supports a measure $\mu$, then convergence in measure, for a sequence of functions, is the same as convergence with respect to the (integral of the) metric you cite. That is: on $\mathrm{Meas}(X)$, define the metric 
$$
d_\mu(f, g) = \int \frac{d(f(x), g(x))} {1+d(f(x), g(x))} d\mu(x) 
$$
Then a sequence of measurable functions converges in measure iff it converges with respect to the metric $d_\mu$. 
This is stated in Tao's book on measure theory, if I recall correctly.